\section*{Introduction
			\subsection{The 2-braid group}
			Rouquier complexes were introduced in \cite{Rou_cat} to study actions of the (generalized)
			braid group $B_W$ on categories. 
			
			These actions appeared in representation theory already in the works of Carlin
			\cite{Car} or Rickard \cite{Rick}, and 
			a precise definition was made by Deligne \cite{Deli}.
			They describe higher symmetries of the categories acted upon and usually give rich information 
			about them. A classical example is the 
			braid action on $\mathcal{D}^{\text{b}}(\mathcal{O})$, the bounded derived category of 
			the category $\mathcal{O}$ of representations of a complex semisimplie Lie algebra $\mathfrak{g}$.
			A geometric counterpart is the action on $\mathcal{D}^{\text{b}}(G/B)$, the bounded derived category of 
			sheaves over the flag variety $G/B$ associated with a reductive algebraic group $G$ and 
			a Borel subgroup $B$.
			
			Rouquier pointed out the interest of the category of self-equivalences 
			induced by such an action. So he introduced the 
			\textit{2-braid group} $\mathscr{B}_W$, that upgrades $B_W$ to a category and serves as a 
			model for this purpose.
			It is a subcategory of $\mathcal{K}^{\mathrm{b}}(\mathscr{H})$, the homotopy category of the \emph{Hecke category} $\mathscr{H}$, and its 
			objects, called \emph{Rouquier complexes}, are all possible tensor products between certain 
			\emph{standard} and \emph{costandard} complexes $F_s$ and $F_s^{-1}$, for $s\in S$. 
			Such products satisfy, up to homotopy, the relations of the braid group 
			so their isomorphism classes
			form at least a quotient of $B_W$. See Remark \ref{rmk_faithful} below for more detail.
			
			This category also plays an important role in algebraic topology.
			Khovanov and Rozansky \cite{KhovRoz} began the construction of link invariants from 
			Rouquier complexes, that was made precise by Khovanov \cite{Khov} as
			a triply graded link homology. The idea is that the homology
			of the complex obtained computing Hochschild (co)homology of the Rouquier complex $F_\omega$
			associated to a braid $\omega$, is an invariant, up to an overall shift, of the link $\overline{\omega}$ obtained by closing $\omega$
			(Rouquier complexes come with a cohomological and a polynomial degree, and Hochschild homology
			provides a third grading).

			When working with Rouquier complexes, it is useful to consider appropriate representatives 
			in the category $\mathcal{C}^{\mathrm{b}}(\mathscr{H})$ of complexes. The natural ones are just the tensor products between 
			the standard and costandard complexes. 
			When the category $\mathscr{H}$ is Krull-Schmidt (see Theorem \ref{thm_soergcat}), any complex 
			admits a \emph{minimal subcomplex}: a homotopy equivalent summand with no
			contractible direct summand.
			In this	paper we find an intermediary reduction for positive (or negative) Rouquier complexes 
			(i.e.\ with only $F_s$'s, or resp.\ $F_s^{-1}$'s) which is 
			available with no restriction on the coefficients. The minimal subcomplex is very hard to 
			find in general, but one can see our result as a first step of such a reduction.
			\subsection{Diagrammatic Hecke category}
			As we mentioned, Rouquier complexes are certain objects of the homotopy 
			category of the Hecke category. The original definition was in terms of Soergel bimodules. 
			In this paper we will use the diagrammatic presentation of the category due to Elias 
			and Williamson \cite{EW}.
			 
			The generating objects, denoted $B_s$, correspond to simple reflections, and are 
			represented as colored points (one color for each $s\in S$). 
			Products between these generators are called \emph{Bott-Samelson} objects. They correspond
			to words and are represented by sequences of points in a line.
			The morphisms between 
			Bott-Samelson objects are described by certain \emph{diagrams} inside the strip
			$\mathbb{R}\times [0,1]$. Namely, these are
			planar graphs obtained from some generating vertices, that connect the sequences of 
			colored points corresponding to the source and the target,
			identified by some relations. 
			This \emph{diagrammatic} Hecke category is available for any Coxeter system, equipped with a \emph{realization} $\mathfrak{h}$ (i.e.\ a 
			finite rank representation over the coefficient ring satisfying certain properties).
			For more detail see \S\,\ref{sec_diagHeck} below. Under certain conditions this category is equivalent to 
			that of Soergel bimodules and has a geometric counterpart, when $W$ is a Weyl group, given by 
			equivariant parity sheaves on the corresponding flag variety.

			Let us pass to the homotopy category. For any $s\in S$, the standard and costandard 
			complexes $F_s$ and $F_{s}^{-1}$ are defined as
			\[
				\begin{tikzcd}[row sep=small,%
								execute at end picture={%
									\def.32cm{.32cm}
									\pic[red,yshift=.3em] at (A) {dot={.32cm}};
									\pic[red,yshift=1.5em,yscale=-1] at (B) {dot={.32cm}};
									}]
					F_{\re{s}}=\,\cdots\ar[r]		& 0 \ar[r]		& 0	\ar[r]								& B_{\re{s}}\ar[r,""{coordinate,name=A}]	& \mathbbm{1}(1)\ar[r]	& 0\ar[r]		& \cdots \\
																& \cohdeg{-2}	& \cohdeg{-1}							& \cohdeg{0}											& \cohdeg{1}	& \cohdeg{2}	& \\
					F_{\re{s}}^{-1}=\,\cdots\ar[r]	& 0\ar[r]		& \mathbbm{1}(-1)\ar[r,""{coordinate,name=B}]	& B_{\re{s}}\ar[r]							& 0	\ar[r]		& 0\ar[r]		& \cdots \\
				\end{tikzcd}
			\]			
			where the numbers in the middle row denote the 
			cohomological degree in $\mathcal{K}^{\mathrm{b}}(\mathscr{H})$. 
			Then Rouquier complexes are are all possible products between the $F_{s}^{\pm 1}$'s.
			\subsection{Results}
			We consider a \emph{positive} Rouquier complex, of the form:
			\[
				F_{\underline{w}}^\bullet=F_{s_1}F_{s_2}\dots F_{s_n}			
			\]
			for $s_i\in S$. As a graded object, this is the direct sum of Bott-Samelson 
			objects indexed by the $2^n$ subexpressions of $\underline{w}=s_1s_2\dots s_n$.
			We find a summand $F_{\underline{w}}$, in the category $\mathcal{C}^{\mathrm{b}}(\mathscr{H})$ which, 
			as a graded object, is a direct sum indexed over \emph{subwords} (each of which can correspond
			to several subexpressions). Then we show the following (see Theorem \ref{thm_reduce} below).
			\begin{thm*}
				The inclusion of the complex $F_{\underline{w}}$ in $\Roudg{\underline{w}}$ is a homotopy equivalence.
			\end{thm*}
			Let us see an example of our simplification.
			\begin{exa}
				Let $\underline{w}=ssttss$. The original complex $\Roudg{\underline{w}}$ is drawn in Figure \ref{fig_6cube}. 
				It is a cube of dimension $6$, whose vertices are labeled by the $2^6=64$ subexpressions of 
				$ssttss$. The arrows (the edges of the cube) describe the differential map: each subexpression
				has a map towards those subexpressions that can be obtained by canceling a symbol from it.
				\begin{figure}[h]
					\centering
					\begin{tikzpicture}[x=1.6cm,y=.2cm]
						\coordinate (v000000) at (0.3,0);
						\coordinate (v000001) at (1,5);\coordinate (v000010) at (1,3);\coordinate (v000100) at (1,1);\coordinate (v001000) at (1,-1);\coordinate (v010000) at (1,-3);\coordinate (v100000) at (1,-5);
						\coordinate (v000011) at (2,14);\coordinate (v000101) at (2,12);\coordinate (v000110) at (2,10);\coordinate (v001001) at (2,8);\coordinate (v001010) at (2,6);\coordinate (v001100) at (2,4);\coordinate (v010001) at (2,2);\coordinate (v010010) at (2,0);\coordinate (v010100) at (2,-2);\coordinate (v011000) at (2,-4);\coordinate (v100001) at (2,-6);\coordinate (v100010) at (2,-8);\coordinate (v100100) at (2,-10);\coordinate (v101000) at (2,-12);\coordinate (v110000) at (2,-14);
						\coordinate (v000111) at (3,19);\coordinate (v001011) at (3,17);\coordinate (v001101) at (3,15);\coordinate (v001110) at (3,13);\coordinate (v010011) at (3,11);\coordinate (v010101) at (3,9);\coordinate (v010110) at (3,7);\coordinate (v011001) at (3,5);\coordinate (v011010) at (3,3);\coordinate (v011100) at (3,1);\coordinate (v100011) at (3,-1);\coordinate (v100101) at (3,-3);\coordinate (v100110) at (3,-5);\coordinate (v101001) at (3,-7);\coordinate (v101010) at (3,-9);\coordinate (v101100) at (3,-11);\coordinate (v110001) at (3,-13);\coordinate (v110010) at (3,-15);\coordinate (v110100) at (3,-17);\coordinate (v111000) at (3,-19);
						\coordinate (v001111) at (4,14);\coordinate (v010111) at (4,12);\coordinate (v011011) at (4,10);\coordinate (v011101) at (4,8);\coordinate (v011110) at (4,6);\coordinate (v100111) at (4,4);\coordinate (v101011) at (4,2);\coordinate (v101101) at (4,0);\coordinate (v101110) at (4,-2);\coordinate (v110011) at (4,-4);\coordinate (v110101) at (4,-6);\coordinate (v110110) at (4,-8);\coordinate (v111001) at (4,-10);\coordinate (v111010) at (4,-12);\coordinate (v111100) at (4,-14);
						\coordinate (v011111) at (5,5);\coordinate (v101111) at (5,3);\coordinate (v110111) at (5,1);\coordinate (v111011) at (5,-1);\coordinate (v111101) at (5,-3);\coordinate (v111110) at (5,-5);
						\coordinate (v111111) at (5.7,0);
						\foreach 100 in {0,1}{
							\foreach \b in {0,1}{
								\foreach \c in {0,1}{
									\foreach .5cm in {0,1}{
										\foreach \e in {0,1}{
											\foreach \f in {0,1}{
													\fill (v100\b\c.5cm\e\f) circle (1pt);												
													\node (n100\b\c.5cm\e\f) at (v100\b\c.5cm\e\f) {};
												}
											}
										}
									}
								}
							}
						\foreach \b in {0,1}{
							\foreach \c in {0,1}{
								\foreach .5cm in {0,1}{
									\foreach \e in {0,1}{
										\foreach \f in {0,1}{
												\draw[-latex,gray] (n0\b\c.5cm\e\f)--(n1\b\c.5cm\e\f);
											}
										}
									}
								}
							}
						\foreach 100 in {0,1}{
								\foreach \c in {0,1}{
									\foreach .5cm in {0,1}{
										\foreach \e in {0,1}{
											\foreach \f in {0,1}{
												\draw[-latex,gray] (n\a0\c.5cm\e\f)--(n\a1\c.5cm\e\f);												
												}
											}
										}
									}
							}
						\foreach 100 in {0,1}{
							\foreach \b in {0,1}{
									\foreach .5cm in {0,1}{
										\foreach \e in {0,1}{
											\foreach \f in {0,1}{
												\draw[-latex,gray] (n100\b0.5cm\e\f)--(n100\b1.5cm\e\f);
												}
											}
										}
								}
							}
						\foreach 100 in {0,1}{
							\foreach \b in {0,1}{
								\foreach \c in {0,1}{
										\foreach \e in {0,1}{
											\foreach \f in {0,1}{
												\draw[-latex,gray] (n100\b\c0\e\f)--(n100\b\c1\e\f);
												}
											}
										}
									}
								}
						\foreach 100 in {0,1}{
							\foreach \b in {0,1}{
								\foreach \c in {0,1}{
									\foreach .5cm in {0,1}{
											\foreach \f in {0,1}{
												\draw[-latex,gray] (n100\b\c\d0\f)--(n100\b\c\d1\f);
												}
											}
										}
									}
								}
						\foreach 100 in {0,1}{
							\foreach \b in {0,1}{
								\foreach \c in {0,1}{
									\foreach .5cm in {0,1}{
										\foreach \e in {0,1}{
												\draw[-latex,gray] (n100\b\c.5cm\e0)--(n100\b\c.5cm\e1);
												}
											}
										}
									}
								}
					\end{tikzpicture}
					\caption{A picture of the complex $\Roudg{\underline{ssttss}}$.}\label{fig_6cube}
				\end{figure}
				The complex $F_{\underline{w}}$ instead is showed in Figure \ref{fig_cpxssttss}. The vertices this time are labeled by the 
				23 subwords and the arrows (representing the new differential) connect a certain word with all its subwords (the symbol $\emptyset$ denotes the empty word).
				\begin{figure}[h]
					\centering
					\begin{tikzpicture}[x=1.3cm,y=.35cm]
						\coordinate (v) at (5.6,0);
						\coordinate (s) at (5,1);\coordinate (t) at (5,-1);
						\coordinate (st) at (4,3);\coordinate (ss) at (4,1);\coordinate (tt) at (4,-1);\coordinate (ts) at (4,-3);
						\coordinate (sst) at (3,5);\coordinate (stt) at (3,3);\coordinate (sss) at (3,1);\coordinate (sts) at (3,-1);\coordinate (tss) at (3,-3);\coordinate (tts) at (3,-5);
						\coordinate (sstt) at (2,5);\coordinate (ssts) at (2,3);\coordinate (ssss) at (2,1);\coordinate (stts) at (2,-1);\coordinate (stss) at (2,-3);\coordinate (ttss) at (2,-5);
						\coordinate (sstss) at (1,0);\coordinate (sstts) at (1,3);\coordinate (sttss) at (1,-3);
						\coordinate (ssttss) at (0,0);
						\node (nv) at (v) {$\emptyset$};
						\foreach \i in {s,t,ss,st,ts,tt,sss,sst,sts,stt,tss,tts,ssss,ssts,sstt,stss,stts,ttss,sstts,sstss,sttss,ssttss}{
							\node (n\i) at (\i) {$\i$};
						}
						\draw[-latex,gray] (nssttss)--(nsstts);\draw[-latex,gray] (nssttss)--(nsstss);\draw[-latex,gray] (nssttss)--(nsttss);
						\draw[-latex,gray] (nsstts)--(nsstt);\draw[-latex,gray] (nsstts)--(nssts);\draw[-latex,gray] (nsstts)--(nstts);
						\draw[-latex,gray] (nsstss)--(nssts);\draw[-latex,gray] (nsstss)--(nssss);\draw[-latex,gray] (nsstss)--(nstss);
						\draw[-latex,gray] (nsttss)--(nstts);\draw[-latex,gray] (nsttss)--(nstss);\draw[-latex,gray] (nsttss)--(nttss);
						\draw[-latex,gray] (nssss)--(nsss);
						\draw[-latex,gray] (nssts)--(nsst);\draw[-latex,gray] (nssts)--(nsss);\draw[-latex,gray] (nssts)--(nsts);
						\draw[-latex,gray] (nsstt)--(nsst);\draw[-latex,gray] (nsstt)--(nstt);
						\draw[-latex,gray] (nstss)--(nsts);\draw[-latex,gray] (nstss)--(nsss);\draw[-latex,gray] (nstss)--(ntss);
						\draw[-latex,gray] (nstts)--(nstt);\draw[-latex,gray] (nstts)--(nsts);\draw[-latex,gray] (nstts)--(ntts);
						\draw[-latex,gray] (nttss)--(ntts);\draw[-latex,gray] (nttss)--(ntss);
						\draw[-latex,gray] (nsss)--(nss);
						\draw[-latex,gray] (nsst)--(nss);\draw[-latex,gray] (nsst)--(nst);
						\draw[-latex,gray] (nsts)--(nst);\draw[-latex,gray] (nsts)--(nss);\draw[-latex,gray] (nsts)--(nts);
						\draw[-latex,gray] (nstt)--(nst);\draw[-latex,gray] (nstt)--(ntt);
						\draw[-latex,gray] (ntss)--(nts);\draw[-latex,gray] (ntss)--(nss);
						\draw[-latex,gray] (ntts)--(ntt);\draw[-latex,gray] (ntts)--(nts);
						\draw[-latex,gray] (nss)--(ns);
						\draw[-latex,gray] (nst)--(ns);\draw[-latex,gray] (nst)--(nt);
						\draw[-latex,gray] (nts)--(nt);\draw[-latex,gray] (nts)--(ns);
						\draw[-latex,gray] (ntt)--(nt);
						\draw[-latex,gray] (ns)--(nv);\draw[-latex,gray] (nt)--(nv);
					\end{tikzpicture}
					\caption{The complex $F_{\underline{ssttss}}$.}\label{fig_cpxssttss}
				\end{figure}
			\end{exa}
			This reduction is obtained via a strong use of \emph{Gaussian elimination for complexes}.
			This is a tool from homological algebra which allows to reduce a given complex when some component
			of the differential is an isomorphism. In \S\,\ref{subs_gausselim}, we develop a 
			``large scale'' version of this technique.
		\section{The diagrammatic Hecke category}\label{sec_diagHeck}
		We now recall the construction, 
		by Elias and Williamson \cite{EW}, of the \emph{diagrammatic Hecke category} associated with 
		(a realization of) an arbitrary Coxeter system.	We begin with some background about Coxeter
		systems, braid groups and Hecke algebras.
		\subsection{Notation for Coxeter systems}
		Let $(W,S)$ be a Coxeter system. For each $s,t\in S$ let $m_{st}$ be the order of $st$ in $W$ (in particular $m_{ss}=1$),
		so that
		\[
			W=\langle s\in S\mid (st)^{m_{st}}=1, \text{ if } m_{st}<\infty \rangle.
		\]
		We will restrict to the case $|S|<\infty$ of finitely generated Coxeter systems. 

		Let $\mathfrak{W}_S$ be the free monoid generated by $S$. Its elements are called \emph{Coxeter words}
		and will be denoted by underlined letters. 
		If $\underline{w}$ is a Coxeter word, we say that it \emph{expresses}
		the element $w\in W$ if $w$ is its image via the natural morphism $\mathfrak{W}_S\rightarrow W$ (this morphism
		is surjective because each $s$ is an involution). 
		We say that a word is \emph{monotonous} if it contains only (repetitions of) one letter.		
		Let $\ell(\underline{w})$ denote the length of $\underline{w}$, i.e.\ the number of its letters.
		The word $\underline{w}$ is said to be \emph{reduced} if there is no shorter word expressing $w$.

		For $\underline{w}, \underline{x} \in \mathfrak{W}_S$, we say that $\underline{x}$ is a \emph{subword} of $\underline{w}$, and we 
		write $\underline{x}\preceq \underline{w}$, 
		if $\underline{x}$ is obtained from $\underline{w}$ by erasing some letters. In other words,
		if $\underline{w}=s_1s_2\cdots s_n$ with $s_i\in S$, then 
		$\underline{x}$ is of the form $s_{i_1}s_{i_2}\cdots s_{i_r}$, 
		with $1\le i_1 \le i_2 \le \dots \le i_r\le n$. A \emph{subexpression} is instead the datum of a subword
		together with the information of the precise positions of the letters in the original word. This 
		can be encoded in a \emph{01-sequence} $\mathbf{e}\in\{0,1\}^n$ where the ones correspond to the letters
		that we pick and the zeros to those we do not pick.
		\begin{exa}
			Let $\underline{w}=s_4s_2s_5s_1s_1s_3$. Then $\underline{x}=s_4s_2s_1$ is a subword, obtained by erasing $s_5$, $s_3$ and one of the two
			$s_1$'s. It corresponds to both subexpressions $110010$ and $110100$.
		\end{exa}		
		
		
		\subsection{Braid groups}
		Let $(W,S)$ be a Coxeter system. We associate to it the corresponding 
		\emph{Artin-Tits group} (or \emph{generalized braid group}):
		\[
			B_W=\langle \sigma_s, s\in S\mid \underbrace{\sigma_s\sigma_t \dots}_{m_{st}}=\underbrace{\sigma_t\sigma_s \dots}_{m_{st}} \rangle.
		\]
		Hence we have a natural surjection $B_W\twoheadrightarrow W$ sending both $\sigma_s$ and 
		$\sigma_s^{-1}$ to $s$. Let $\Sigma^+=\{\sigma_s\}_{s\in S}$,
		$\Sigma^-=\{\sigma_s^{-1}\}_{s\in S}$ and $\Sigma=\Sigma^+ \sqcup \Sigma^-$. 
		Let $\mathfrak{W}_{\Sigma}$ be the free monoid generated by $\Sigma$. Its elements are called 
		\emph{braid words}. We call the elements of $\Sigma^+$ (and $\Sigma^-$) \emph{positive letters}
		(respectively \emph{negative letters}).
		A braid word is called \emph{positive} (or \emph{negative}) if all its letters are.
		Subwords and subexpressions are defined in $\mathfrak{W}_{\Sigma}$ in the same way as in $\mathfrak{W}_S$.

		We call \emph{Coxeter projection} the map $\mathfrak{W}_{\Sigma}\rightarrow\mathfrak{W}_S$, sending both 
		$\sigma_s$ and $\sigma_s^{-1}$ to $s$. This has two distinguished sections: any Coxeter word $\underline{w}=s_1s_2\dots s_k$
		has a \emph{positive word lift} $\sigma_{s_1}\sigma_{s_2}\dots \sigma_{s_k}$
		and a \emph{negative word lift} $\sigma_{s_1}^{-1}\sigma_{s_2}^{-1}\dots \sigma_{s_k}^{-1}$. 
		
		Given a non-empty $\underline{\omega}\in\mathfrak{W}_{\Sigma}$, we can write it as
		\begin{equation}\label{eq_posnegdec}
			\underline{\omega}_1\underline{\omega}_2\dots \underline{\omega}_k		
		\end{equation}
		where $\underline{\omega}_i$ are non-empty words such that: each $\underline{\omega}_i$ is either positive or negative;
		$\underline{\omega}_i$ is positive if and only if $\underline{\omega}_{i+1}$ is negative.
		We call \eqref{eq_posnegdec} the \emph{alternating} decomposition of $\underline{\omega}$.

		Finally let $\varpi:\mathfrak{W}_{\Sigma}\rightarrow (\mathbb{Z},+)$ be the morphism of monoids with
		$\varpi(\sigma_s)=1$ and $\varpi(\sigma_s^{-1})=-1$.
		Hence $\varpi$ counts the difference between the number of positive and negative letters.
		\subsection{The Hecke algebra}\label{subs_Heckealg}
		Let $(W,S)$ be a Coxeter system and consider the ring $\ZZ[v,v^{-1}]$ of Laurent polynomial in one 
		variable with integer coefficients.
		\begin{dfn}
			The \emph{Hecke algebra} $\mathbf{H}$ associated with $(W,S)$ is the $\ZZ[v,v^{-1}]$-algebra generated by $\{\delta_s\}_{s\in S}$
			 with relations 
			\[
				\begin{cases}
					(\delta_s+v)(\delta_s-v^{-1})=0& 	\text{for all $s\in S$}\\
					\underbrace{\delta_s\delta_t\dots}_{m_{st}}=\underbrace{\delta_t\delta_s\dots}_{m_{st}}& 		\text{if $m_{st}<\infty$}
				\end{cases}
			\]			
		\end{dfn}		 		
		Notice that $\mathbf{H}$ can be seen as the quotient of the group algebra of the braid group
		$B_W$ over $\ZZ[v,v^{-1}]$, by the above $v$-deformed version of the 
		involution relation in $W$. 
		
		For each $s\in S$ let $b_s:=\delta_s+v$. By the first relation, one gets
		\begin{equation}\label{eq_bsbsalgebra}
			b_sb_s=vb_s+v^{-1}b_s
		\end{equation}
		\begin{rmk}
			Recall that $\mathbf{H}$ is a free $\ZZ[v,v^{-1}]$-module with a 
			\emph{standard} basis $\{\delta_w\}_{w\in W}$, where 
			$\delta_w:=\delta_{s_1}\delta_{s_2}\cdots \delta_{s_k}$ for any reduced 
			expression $\underline{w}=s_1 s_2 \cdots s_k$ for $w$ (one shows that it does not depend on the chosen expression). One can actually start from this free module and 
			define the algebra structure in terms of the standard basis, see \cite[\S\,6]{BjoBre}, 
			\cite[IV, \S\,2, Exercice 23]{Bour}, or \cite[\S\,7]{Humphr_refl}.		
		\end{rmk}
		The category that we are going to describe is a \emph{categorification} 
		of the Hecke algebra. We need to introduce the notion
		of \emph{realization} of a Coxeter system.
		\subsection{Realizations of Coxeter systems}\label{sec_realCox} Let $(W,S)$ be a Coxeter system
		and $\Bbbk$ a commutative ring. A \emph{realization}
		of $W$, in the sense of \cite[\S\,3.1]{EW}, consists in a free,
		finite rank, $\Bbbk$-module $\mathfrak{h}$, with distinguished elements 
		$\{\alpha_s^\vee\}_{s\in S} \subset \mathfrak{h}$ and 
		$\{ \alpha_s \}_{s\in S} \subset \mathfrak{h}^*=\Hom(\mathfrak{h},k)$,
		that we call respectively \emph{simple coroots} and \emph{simple roots}, 
		satisfying the following conditions.
		\begin{enumerate}
		\item If $\langle \cdot\,,\cdot\rangle:\mathfrak{h}^*\times \mathfrak{h}\rightarrow \Bbbk$ is the natural
			pairing, then $\langle \alpha_s,\alpha_s^\vee\rangle =2$, for each $s\in S$.
		\item\label{item_cond2} The $\Bbbk$-module $\mathfrak{h}$ is a representation of $W$ via 
			\[
				s(v)=v-\langle \alpha_s, v\rangle \alpha_s^{\vee},\quad\forall v\in \mathfrak{h},\,\forall s \in S.		
			\]
			Then $W$ acts on $\mathfrak{h}^*$ by the contragredient representation, described by similar formulas:
			\begin{equation}\label{eq_realiz2}
				s(\lambda)=\lambda-\langle \lambda, \alpha_s^\vee \rangle \alpha_s,\quad\quad\forall \lambda\in \mathfrak{h}^*,\,\forall s \in S.		
			\end{equation}		
		\item The technical condition \cite[\S\,3.1, (3.3)]{EW} is satisfied.
		\end{enumerate}


		The action of the realization over simple roots and coroots is encoded in the coefficients\footnote{For 
		the notation, we follow Bourbaki \cite{Bour}: notice that the pairing considered here is the transposed
		of that of \cite{EW}, and \cite{Kac}.}
		$a_{st}:=\langle \alpha_s,\alpha_t^\vee\rangle$, that we store in the 
		\emph{Cartan matrix} $( a_{st} )_{s,t\in S}$.

		Let $R=S(\mathfrak{h}^*)$, with $\mathfrak{h}^*$ in degree 2. The action of $W$ on 
		$\mathfrak{h}^*$ defined by \eqref{eq_realiz2} extends naturally to $R$. 
		We define, for each $s\in S$, the \emph{Demazure operator} $\partial_s : R \rightarrow R$, via:
		\[
						f \mapsto \frac{f-s(f)}{\alpha_s}.
		\]
		Then, in particular, we have\footnote{%
			Notice that $\partial_s(\alpha_t)$ is the same as in \cite{EW}, so this notation avoids 
			transposition issues.}
		$\partial_s(\alpha_t)=\langle \alpha_t,\alpha_s^\vee\rangle$.
		
		In this paper we will only consider \emph{balanced} realizations (see \cite[Definition 3.7]{EW}).
		We also assume \emph{Demazure surjectivity} (see \cite[Assumption 3.9]{EW}), namely we suppose 
		that, for each $s\in S$, the maps:
		\begin{align*}
			&\langle \alpha_s,\cdot \rangle: \mathfrak{h}\rightarrow \Bbbk&
			&\text{and}&
			&\langle \cdot\,,\alpha_s^\vee \rangle: \mathfrak{h}^*\rightarrow \Bbbk
		\end{align*}			
		are surjective. In this case we can (and do) choose some $\delta_s\in\mathfrak{h}^*$ such that 
		$\langle \delta_s,\alpha_s^\vee\rangle=1$.
		Notice that if $2$ is invertible in $\Bbbk$ then this assumption always holds (take for instance
		$\delta_s=\alpha_s/2$). 
		For more details, see \cite[\S\,3.1]{EW}.
		\begin{exa}\label{exa_realiz}
			Here are some interesting examples of balanced realizations
			(cf.\ \cite[Ex.\ 3.3]{EW} and \cite{RW}).
			\begin{enumerate}
				\item\label{item_exgeom} For any Coxeter group $(W,S)$ one can consider the \emph{geometric representation} 
					$\mathfrak{h}:=\bigoplus_{s\in S}\mathbb{R}\alpha_s^\vee$ with 
					$\alpha_s\in \mathfrak{h}^*$ defined via $\langle \alpha_s,\alpha_t^\vee\rangle=-2\cos(\pi/m_{st})$,
					with $m_{ss}:=1$ as usual, and $\pi/\infty:=0$. 
					Then the simple roots are linearly independent if and only if $W$ is finite.
				\item\label{item_exWf} If $W_\mathrm{f}$ is the Weyl group associated with a root datum
					$(X,\ZZ\Phi,X^\vee,\ZZ\Phi^\vee)$, as defined in \cite[\S 7]{Spring}, then one has a natural realization coming from the root datum. 
					One takes $\mathfrak{h}^*=\Bbbk \otimes X$ (so $\mathfrak{h}=\Bbbk \otimes X^\vee$), and
					chooses sets $\{\alpha_s\}$ and $\{\alpha_s^\vee\}$ of simple roots and coroots respectively (in the usual sense). 
					One can also take $\mathfrak{h}$ to be $\Bbbk\otimes \ZZ\Phi^\vee$ (then a natural 
					assumption to make is that $\Bbbk\otimes \ZZ\Phi \times \Bbbk\otimes \ZZ\Phi^\vee\rightarrow \Bbbk$ 
					is a perfect pairing so as to identify $\mathfrak{h}^*$ with the base-changed root lattice). 
				\item\label{item_exW} For the affine case $W=W_\mathrm{f}\ltimes \ZZ\Phi^\vee$, if $s\in S\setminus S_{\mathrm{f}}$ is an
					affine simple reflection, consider the image of $s$ under the projection $W\rightarrow W_{\mathrm{f}}$. This will be a
					reflection associated to some positive coroot $\tilde{\alpha}^\vee$. Then we put 
					$\alpha_{s}^\vee:=-\tilde{\alpha}^\vee$ and $\alpha_{s}:=-\tilde{\alpha}$.
					In this case neither $\{\alpha_s\}$, nor $\{\alpha_s^\vee\}$ are linearly independent.
					This is the realization considered in
					\cite{RW}, up to a different choice of notation (exchanging roots and coroots). 
					Notice that the action of $W$ here factors through that of $W_{\mathrm{f}}$.
				\item\label{item_exKac} One can start with a \emph{generalized Cartan matrix}
					$(a_{st})_{s,t\in S}$ (i.e.\ a matrix satisfying $a_{ss}=2$, $a_{st}\in\mathbb{Z}_{\le 0}$ and 
					$a_{st}=0\Leftrightarrow a_{ts}=0$)
					and build a realization as described in \cite{Kac}. 
					One considers the Kac-Moody Lie algebra $\mathfrak{g}$, defined by $(a_{st})$, 
					and takes $\mathfrak{h}$ to be any $\mathbb{Z}$-lattice of the Cartan subalgebra of $\mathfrak{g}$, 
					containing the coroot lattice and such that the dual lattice $\mathfrak{h}^*$ contains the root 
					lattice. In this way one obtains the \emph{Kac-Moody} realization of the Weyl group
					$W$ associated with $\mathfrak{g}$.
					Here both the simple roots and the simple coroots
					are linearly independent.
				 \item\label{item_exKac2} More generally,
				 	one can consider any \emph{Kac-Moody root datum} associated to
				 	a generalized Cartan matrix as above. This is a free $\mathbb{Z}$-module of finite rank $\Lambda$
				 	with distinguished elements $\{\alpha_s\}\subset \Lambda$ and $\{\alpha_s^\vee\}\subset \Hom(\Lambda,\mathbb{Z})$,
				 	such that $\alpha_s^\vee(\alpha_t)=a_{ts}$.	Then one can take $\mathfrak{h}^*=\Bbbk\otimes \Lambda$ and
				 	$\mathfrak{h}=\Bbbk\otimes \Hom(\Lambda,\mathbb{Z})$.
				 	From the same data one can construct a Kac-Moody ind-group scheme $\mathscr{G}$ over $\mathbb{Z}$, with
				 	canonical Borel subgroup and a maximal torus, such that the character group of the 
				 	latter identifies with $\Lambda$. Then $\mathfrak{h}$ is a realization of the Weyl group associated to $\mathscr{G}$.
				 	Hence this example is a generalization of \ref{item_exWf}.
			\end{enumerate}
		\end{exa}
		\subsection{Definition of the category}\label{subs_defcatHeck}
		Given a realization $\mathfrak{h}$ over $\Bbbk$ of a Coxeter system $(W,S)$, 
		one constructs the corresponding diagrammatic Hecke category $\mathscr{H}=\mathscr{H}(\mathfrak{h},\Bbbk)$.
		This is a $\Bbbk$-linear monoidal category enriched in graded $R$-bimodules.
		First one defines the Bott-Samelson category $\mathscr{H}_{\mathrm{BS}}$ by generators and relations,
		then one gets $\mathscr{H}$ as the Karoubi envelope of the closure of $\mathscr{H}_{\mathrm{BS}}$ by direct 
		sums and shifts.	
		\begin{enumerate}
			\item The objects of $\D_{\mathrm{BS}}$ are generated by tensor product from objects $B_s$ for $s\in S$. So a general object
				corresponds to a Coxeter word: if
				$\underline{w}=s_1\dots s_n$, let $B_{\underline{w}}$ denote the object
				$B_{s_1}\otimes \cdots \otimes B_{s_n}$. Let also $\mathbbm{1}$ denote the monoidal unit, corresponding to the 
				empty word. 
			\item Morphisms in $\Hom_{\D_{\mathrm{BS}}}(B_{\underline{w}_1},B_{\underline{w}_2})$
				are $\Bbbk$-linear combinations of \emph{Soergel graphs}, which are defined as follows.
				\begin{itemize}
					\item We associate a color to each simple reflection.
					\item A Soergel graph is then a colored, \emph{decorated} planar graph 
						contained in the planar strip $\mathbb{R}\times [0,1]$, with boundary in 
						$\mathbb{R}\times\{0,1\}$.
					\item The bottom (and top) boundary is the arrangement
						of boundary points colored according to the letters of the source word $\underline{w}_1$ 
						(and target word $\underline{w}_2$, respectively).
					\item The edges of the graph are colored in such a way that those connected with the 
						boundary have colors consistent with the boundary points.
					\item The other vertices of the graph are either: 
						\begin{enumerate}
							\item univalent (called \emph{dots}), which are declared of degree 1, or;
							\item trivalent	with three edges of the same color, of degree $-1$, or;
							\item $2m_{st}$-valent with edges of alternating colors corresponding to $s$ and $t$,
							 	if $m_{st}<\infty$, of degree 0.
								By poetic licence, we also call them $(s,t)$-ars.
						\end{enumerate}
						\begin{center}
							\begin{tikzpicture}[baseline=0,scale=.7]
								\def4cm{4cm}
								\draw[gray,dashed] (0,0) circle (1cm); %
									\draw[red] (0,-1)--(0,0); \fill[red] (0,0) circle (2pt);
								\node[anchor=north,inner sep=8pt] at (0,-1) {(a) \textit{Dot}};
								\begin{scope}[xshift=4cm]
									\draw[gray,dashed] (0,0) circle (1cm); %
										\draw[red] (0,1)--(0,0); %
											\draw[red] (-30:1cm)--(0,0)--(-150:1cm);
									\node[anchor=north,inner sep=8pt] at (0,-1) {(b) \textit{Trivalent}};
								\end{scope}
								\begin{scope}[xshift=2*4cm,rotate=-15]
									\draw[gray,dashed] (0,0) circle (1cm); %
										\draw[red] (0,1)--(0,0); %
											\draw[red] (-30:1cm)--(150:1cm);%
												\draw[red](-150:1cm)--(30:1cm);
										\draw[blue,rotate=30] (0,1)--(0,0); %
											\draw[blue,rotate=30] (-30:1cm)--(150:1cm);%
												\draw[blue,rotate=30](-150:1cm)--(30:1cm);
									\foreach \i in {-60,-75,-90}{%
											\fill (\i:.5cm) circle (.5pt);
										}
								\end{scope}
								\node[anchor=north,xshift=1.4*4cm,inner sep=8pt] at (0,-1) {(c) \textit{$(s,t)$-ar}};
							\end{tikzpicture}
						\end{center}
					\item Decorations are boxes labeled by homogeneous elements in 
						$R$ that can appear in
						any region (i.e.\ connected component of the complement of the graph): 
						we will usually omit the boxes and just write the polynomials.
				\end{itemize}
				Then, composition of morphism is given by gluing diagrams vertically, whereas tensor product
				is given by gluing them horizontally. The identity morphism of the object $B_{\underline{w}}$ is
				the diagram with parallel vertical strands colored according to the word $\underline{w}$.
			\item These diagrams are identified via some relations:
				\begin{description}
					\item[Polynomial relations.] The first such relation says that whenever two polynomials are in the same region, they multiply.
						This gives morphism spaces the structure of $R$-bimodules, by acting on the leftmost or the 
						rightmost region. 
						
						The other polynomial relations are:
						\begin{align*}
							& \text{\emph{Barbell} relation:}  &
							\begin{tikzpicture}[baseline=-0.1cm,scale=.7, transform shape] 
								\draw[gray,dashed] (0,0) circle (1cm); \draw[red] (0,.5)--(0,-.5); \fill[red] (0,.5) circle (2pt);\fill[red] (0,-.5) circle (2pt);
							\end{tikzpicture}
							&=
							\begin{tikzpicture}[baseline=-0.1cm,scale=.7, transform shape]
								\draw[gray,dashed] (0,0) circle (1cm); \node[draw] at (0,0) {$\alpha_{\re{s}}$};
							\end{tikzpicture}			\\
							& \text{\emph{Sliding} relation:} & 
							\begin{tikzpicture}[baseline=-0.1cm,scale=.7, transform shape]
								\draw[gray,dashed] (0,0) circle (1cm); \draw[red] (0,1)--(0,-1); 
								\node[draw] at (.5,0) {$f$};
							\end{tikzpicture}			
							&=
							\begin{tikzpicture}[baseline=-0.1cm,scale=.7, transform shape]
								\draw[gray,dashed] (0,0) circle (1cm); \draw[red] (0,1)--(0,-1); 
								\node[draw, inner sep =.08cm] at (-.5,0) {$\re{s}(f)$};
							\end{tikzpicture}			
							+
							\begin{tikzpicture}[baseline=-0.1cm,scale=.7, transform shape]
								\draw[gray,dashed] (0,0) circle (1cm); \draw[red] (0,1)--(0,.5); \fill[red] (0,.5) circle (2pt);
								\draw[red] (0,-1)--(0,-.5); \fill[red] (0,-.5) circle (2pt); 
								\node[draw] at (0,0) {$\partial_{\re{s}}(f)$};
							\end{tikzpicture}
							\end{align*}
					\item[One color relations.] These are the following:
						\begin{align}
							& \text{\emph{Frobenius associativity}:} &
							\begin{tikzpicture}[baseline=-0.1cm,color=red,scale=sqrt(2)/3,scale=.7, transform shape]\label{associativity}
								\draw (-1.5,-1.5)--(-0.6,0)--(0.6,0)--(1.5,-1.5);
								\draw (-1.5,1.5)--(-0.6,0); \draw(0.6,0)--(1.5,1.5);
								\draw[dashed,gray] (0,0) circle ({3/sqrt(2)});
							\end{tikzpicture}
							&=
							\begin{tikzpicture}[baseline=-0.1cm,color=red,scale=sqrt(2)/3,rotate=90,scale=.7, transform shape]
								\draw (-1.5,-1.5)--(-0.6,0)--(0.6,0)--(1.5,-1.5);
								\draw (-1.5,1.5)--(-0.6,0); \draw(0.6,0)--(1.5,1.5);
								\draw[dashed,gray] (0,0) circle ({3/sqrt(2)});
							\end{tikzpicture} \\
							& \text{\emph{Frobenius unit}:} &
							\begin{tikzpicture}[color=red,baseline=-0.1cm,scale=.7, transform shape]\label{dotline}
								\draw[gray,dashed] (0,0) circle (1cm);
								\draw (0,-1) -- (0,1); \draw (0,0)--(.3,0);\fill (0.3,0) circle (2pt);
							\end{tikzpicture}
							&=
							\begin{tikzpicture}[baseline=-0.1cm,scale=.7, transform shape]
								\draw[gray,dashed] (0,0) circle (1cm);						
								\draw[red] (0,-1) -- (0,1);
							\end{tikzpicture}
							\\
							&\text{\emph{Needle} relation:} &
							\begin{tikzpicture}[baseline=0,scale=.7, transform shape]\label{eq_needle}
								\draw[gray,dashed] (0,0) circle (1cm);
								\draw[red] (0,-1)--(0,-0.3) arc (-90:270:0.3cm);
							\end{tikzpicture}
							&= 0 \notag
						\end{align}
						\begin{rmk}\label{rmk_selfbiadj}				
							Relations \eqref{associativity} and \eqref{dotline} could be phrased by saying that
							$B_s$ is a \emph{Frobenius algebra object} in the category $\D_{\mathrm{BS}}$, which explains the terminology. 
							This was pointed out in 
							\cite{EliKho}. See \cite[\S\,8]{EMTW}.
						\end{rmk}
					\item[Two color relations.] These allow to move dots,
						or trivalent vertices, across $(s,t)$-ars.
						We give the two versions, according to the parity of $m_{st}$:
						\[
							\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
								\draw[dashed,gray] (0,0) circle (1cm);
								\coordinate (a) at (-.3,0); \coordinate (b) at (0.5,0);
								\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(a);
								\draw[red] ({180-2*360/11}:1cm)--(a); \draw[blue] ({180-5*360/11}:1cm)--(b);
								\draw[red] ({180-4*360/11}:1cm)--(a)
								\draw[blue] ({180+360/11}:1cm)--(a);
								\draw[red] ({180+2*360/11}:1cm)--(a); \draw[blue] ({180+5*360/11}:1cm)--(b);
								\draw[red] ({180+4*360/11}:1cm)--(a)
								\draw[blue](a)--(b);
								\node at (-0.05,.5) {...};\node at (-0.05,-.5) {...};
							\end{tikzpicture}
							=
							\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
								\draw[dashed,gray] (0,0) circle (1cm);
								\coordinate (a) at (-.5,0); \coordinate (b) at (0.15,0.35); \coordinate (c) at (0.15,-0.35);
								\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(b);
								\draw[red] ({180-2*360/11}:1cm)--(b); \draw[blue] ({180-5*360/11}:1cm)--(b);
								\draw[red] ({180-4*360/11}:1cm)--(b)
								\draw[blue] ({180+360/11}:1cm)--(c);
								\draw[red] ({180+2*360/11}:1cm)--(c); \draw[blue] ({180+5*360/11}:1cm)--(c);
								\draw[red] ({180+4*360/11}:1cm)--(c)
								\draw[blue](c)..controls (-0.15,0)..(b); \draw[red] (b) -- (c);\draw[red] (b)--(a)--(c);
								\draw[red] (b) ..controls (0.7,0).. (c);
								\node at (.2,.6) {...};\node at (.2,-.6) {...};\node at (.35,0) {...};
							\end{tikzpicture}\quad\text{or}\quad
							\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
								\draw[dashed,gray] (0,0) circle (1cm);
								\coordinate (a) at (-.3,0); \coordinate (b) at (0.5,0);
								\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(a);
								\draw[red] ({180-2*360/11}:1cm)--(a); \draw[red] ({180-5*360/11}:1cm)--(b);
								\draw[blue] ({180-4*360/11}:1cm)--(a)
								\draw[blue] ({180+360/11}:1cm)--(a);
								\draw[red] ({180+2*360/11}:1cm)--(a); \draw[red] ({180+5*360/11}:1cm)--(b);
								\draw[blue] ({180+4*360/11}:1cm)--(a)
								\draw[red](a)--(b);
								\node at (-0.05,.5) {...};\node at (-0.05,-.5) {...};
							\end{tikzpicture}
							=
							\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
								\draw[dashed,gray] (0,0) circle (1cm);
								\coordinate (a) at (-.5,0); \coordinate (b) at (0.15,0.35); \coordinate (c) at (0.15,-0.35);
								\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(b);
								\draw[red] ({180-2*360/11}:1cm)--(b); \draw[red] ({180-5*360/11}:1cm)--(b);
								\draw[blue] ({180-4*360/11}:1cm)--(b)
								\draw[blue] ({180+360/11}:1cm)--(c);
								\draw[red] ({180+2*360/11}:1cm)--(c); \draw[red] ({180+5*360/11}:1cm)--(c);
								\draw[blue] ({180+4*360/11}:1cm)--(c)
								\draw[blue](c)..controls (-0.15,0)..(b); \draw[red] (b) -- (c);\draw[red] (b)--(a)--(c);
								\draw[blue] (b) ..controls (0.7,0).. (c);
								\node at (.2,.6) {...};\node at (.2,-.6) {...};\node at (.35,0) {...};
							\end{tikzpicture}
						\]
						and
						\[
							\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
								\draw[dashed,gray] (0,0) circle (1cm);
								\coordinate (a) at (-.2,0); \coordinate (b) at (0.5,0);
								\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(a);
								\draw[blue] ({180-3*360/11}:1cm)--(a); 
								\draw[red] ({180-4*360/11}:1cm)--(a)
								\draw[blue] ({180+360/11}:1cm)--(a);
								\draw[blue] ({180+3*360/11}:1cm)--(a); \fill[blue] (b) circle (2pt);
								\draw[red] ({180+4*360/11}:1cm)--(a)
								\draw[blue](a)--(b);
								\node at (-0.4,.5) {...};\node at (-0.4,-.5) {...};
							\end{tikzpicture}
							=
							\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
								\draw[dashed,gray] (0,0) circle (1cm);
								\coordinate (a) at (-.2,0); \coordinate (b) at (0.5,0);
								\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(a);
								\draw[blue] ({180-3*360/11}:1cm)--(a); 
								\draw[blue] ({180+360/11}:1cm)--(a);
								\draw[blue] ({180+3*360/11}:1cm)--(a);
								\draw[red] ({180+4*360/11}:1cm) -- (b) -- ({180-4*360/11}:1cm); \draw[red](a)--(b)
								\node at (-0.4,.6) {...};\node at (-0.4,-.6) {...};
								\node[draw,circle, inner sep=.1cm,fill=white] at (a) {$\JW$};
							\end{tikzpicture}
							\quad\text{or}\quad
							\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
								\draw[dashed,gray] (0,0) circle (1cm);
								\coordinate (a) at (-.2,0); \coordinate (b) at (0.5,0);
								\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(a);
								\draw[red] ({180-3*360/11}:1cm)--(a); 
								\draw[blue] ({180-4*360/11}:1cm)--(a)
								\draw[blue] ({180+360/11}:1cm)--(a);
								\draw[red] ({180+3*360/11}:1cm)--(a); \fill[red] (b) circle (2pt);
								\draw[blue] ({180+4*360/11}:1cm)--(a)
								\draw[red](a)--(b);
								\node at (-0.4,.5) {...};\node at (-0.4,-.5) {...};
							\end{tikzpicture}
							=
							\begin{tikzpicture}[baseline=0,scale=.8,transform shape]
								\draw[dashed,gray] (0,0) circle (1cm);
								\coordinate (a) at (-.2,0); \coordinate (b) at (0.5,0);
								\draw[red] (180:1cm)--(a); \draw[blue] ({180-360/11}:1cm)--(a);
								\draw[red] ({180-3*360/11}:1cm)--(a); 
								\draw[blue] ({180+360/11}:1cm)--(a);
								\draw[red] ({180+3*360/11}:1cm)--(a);
								\draw[blue] ({180+4*360/11}:1cm) -- (b) -- ({180-4*360/11}:1cm); \draw[blue](a)--(b)
								\node at (-0.4,.6) {...};\node at (-0.4,-.6) {...};
								\node[draw,circle, inner sep=.1cm,fill=white] at (a) {$\JW$};
							\end{tikzpicture}
						\]
						where the circles labeled $\JW$ are the \emph{Jones-Wenzel morphisms}. 
						These are certain $\Bbbk$-linear combinations of diagrams (with circular boundary
						and $2m_{st}-2$ boundary points around it)
						that can be described in terms of the 2-colored Temperley-Lieb category
						We will not need these relations in this paper. For further detail, we refer the reader to 
						\cite[\S\,5.2]{EW}, \cite{Elias_two} or \cite[\S\,8]{EMTW}.
					\item[Three color relations.] For each finite parabolic subgroup $W_I$
						of rank 3, there is a relation ensuring compatibility between the three 
						corresponding $2m$-valent vertices. This is an analog of the 
						\emph{Zamolodchikov tetrahedron equation} for braided monoidal 
						2-categories, generalized to all types (not just $A$). 
						We will not need them either, so for more detail see \cite[\S\,1.4.3]{EW}.
				\end{description}				
				Notice that all the relations are homogeneous, so the morphism spaces are \emph{graded}
				$R$-bimodules.
		\end{enumerate}
		This completes the definition of $\D_{\mathrm{BS}}$.
		\begin{dfn*}
			The \emph{diagrammatic Hecke category} $\mathscr{H}$ is the Karoubi envelope of the closure of $\D_{\mathrm{BS}}$
			by shifts, denoted by $(\cdot)$, and direct sums.
		\end{dfn*}
		\subsection{Main properties of the Hecke category}\label{subs_catofHeck}
		The split Grothendieck group $[\mathscr{H}]_{\oplus}$ is naturally a $\ZZ[v,v^{-1}]$-algebra: the ring structure is induced by the tensor product
		and the action of $v$ corresponds to the shift (more precisely $v[B]:=[B(1)]$).
		Hence we can state the following result (see \cite[\S\,6.6]{EW}).
		\begin{thm}\label{thm_soergcat}
			If $\Bbbk$ is a complete local ring, then $\mathscr{H}$ is Krull-Schmidt and 
			there is a unique isomorphism of $\ZZ[v,v^{-1}]$-algebras, called \emph{character},
					\[
						\mathrm{ch}:[\mathscr{H}]\overset{\sim}{\rightarrow}\mathbf{H}
					\]
					sending the class of $B_s$ to $b_s$;
		\end{thm}
		\begin{rmk}
			\begin{enumerate}
				\item Under the hypotheses of Theorem \ref{thm_soergcat} one can also classify the
					indecomposable objects: they are parametrized, up to isomorphism and shift, by $W$. 
					Moreover morphism spaces are free as left $R$-modules and one can express 
					their graded rank in term of their classes in $\mathbf{H}$.
				\item If $\mathrm{char}(\Bbbk)\neq 2$ and that the realization 
					$\mathfrak{h}$ is \emph{reflection faithful} in the sense of \cite[Definition 3.10]{EW},
					then $\mathscr{H}$ is equivalent to the category of Soergel bimodules, see \cite[Theorem\ 6.30]{EW}. 
					This assumption can be dropped by using the variant of the category of 
					Soergel bimodules introduced by Abe \cite{Abe}.
				\item\label{item_geom} Let $G$ be a reductive algebraic group over the complex numbers with
					a Borel subgroup $B$ and a maximal torus $T$.
					Then one can consider the realization of the Weyl group of Example \ref{exa_realiz} \ref{item_exWf}.
					Riche and Williamson \cite{RW} showed that, if $\Bbbk$ is a complete local ring (with 2 invertible in 
					some cases) the category $\mathscr{H}$ is equivalent to that of $B$-equivariant parity sheaves
					over the flag variety $G/B$ with coefficients in $\Bbbk$, in the sense of \cite{JMW},
					endowed with its convolution operation.
					One also has a version for the affine Weyl group by considering the affine flag variety 
					$G(\!(t)\!)/I$ and the realization from Example \ref{exa_realiz} \ref{item_exW} (see \cite[Theorem\ 10.16]{RW}),
					and more generally for any cristallographic group using the Kac-Moody flag variety and the realization in Example 
					\ref{exa_realiz} \ref{item_exKac2} (see \cite[Theorem\ 10.5]{RW}).	
					If $\mathrm{char}(\Bbbk)=0$, then, thanks to the Decomposition Theorem, this category conincides
					with that of semisimple complexes.
	\end{enumerate}
		\begin{exa}\label{exa_basicdeco}
			The basic example of a diagrammatic computation in $\mathscr{H}$ is the following (see \cite[\S\,5]{EW}). 
			If $s\in S$, then
			\begin{equation}\label{eq_BsBscat}
				B_s\otimes B_s\cong B_s(-1)\oplus B_s(1)			
			\end{equation}
			In fact consider the following maps (where we omit the bottom and top lines):
			\[
			\def.5cm{1cm}
				\iota_1=	\begin{tikzpicture}[yscale=-1,baseline=-.5*1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866]
							\draw[red] (.5cm,0) -- (1.5*.5cm,.5*1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866) -- (2*.5cm,0);
							\draw[red] (1.5*.5cm,.5*1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866)--(1.5*.5cm,1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866);
							\node at (1.5*.5cm,0) {$\delta_s$};
						\end{tikzpicture},
						\qquad
				\pi_1=	\begin{tikzpicture}[baseline=.5*1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866]
							\draw[red] (.5cm,0)--(1.5*.5cm,.5*1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866)--(2*.5cm,0);
							\draw[red] (1.5*.5cm,.5*1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866)--(1.5*.5cm,1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866);
						\end{tikzpicture},
						\qquad
				\iota_2=	\begin{tikzpicture}[baseline=-.5*1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866,yscale=-1]
							\draw[red] (.5cm,0)--(1.5*.5cm,.5*1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866)--(2*.5cm,0);
							\draw[red] (1.5*.5cm,.5*1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866)--(1.5*.5cm,1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866);
						\end{tikzpicture},
						\qquad
				\pi_2=	-\begin{tikzpicture}[baseline=.5*1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866]
							\draw[red] (.5cm,0) -- (1.5*.5cm,.5*1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866) -- (2*.5cm,0);
							\draw[red] (1.5*.5cm,.5*1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866)--(1.5*.5cm,1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866);
							\node[yshift=-1pt] at (1.5*.5cm,0) {$s(\delta_s)$};
						\end{tikzpicture}.
			\def.5cm{.5cm}
			\]
			Using the relations, one can check that $\iota_1\pi_1+\iota_2\pi_2=\id_{B_sB_s}$, as well as
			\begin{align*}
				&\pi_1\iota_1=\id_{B_s(-1)},&	 &\pi_2\iota_2=\id_{B_s(1)},&	&\pi_i\iota_j=0\text{ if }i\neq j.
			\end{align*}
			The decomposition \eqref{eq_BsBscat} lifts the equality \eqref{eq_bsbsalgebra} in the Hecke algebra.
		\end{exa}
		\end{rmk}
		\begin{rmk}\label{rmk_onSoergelcat}
			The Hecke category plays an important role in representation theory. Let us mention some examples.
			\begin{enumerate}
				\item\label{item_catO} Let $\mathfrak{g}$ be a complex semisimple Lie algebra and fix a Cartan subalgebra $\mathfrak{h}$. 
					Let $W$ be the corresponding Weyl group and $S$ the set of simple reflections corresponding to a 
					choice of simple roots. Consider \emph{dot action} of $W$ on $\mathfrak{h}^*$ given 
					by $w\cdot \lambda=w(\lambda+\rho)-\rho$ where $\rho$ is half the sum of positive 
					roots. Consider the category $\mathcal{O}$ of Bernstein-Gel'fand-Gel'fand and its 
					decomposition into blocks $\mathcal{O}=\bigoplus\mathcal{O}_\lambda$. For each 
					$s\in S$ let $\mu\in \mathfrak{h}^*$ be a weight with stabilizer $\{1,s\}$ under 
					the dot action. Recall that one has translation functors 
					$T^s:=T_0^{\mu}:\mathcal{O}_0\rightarrow\mathcal{O}_{\mu}$ and 
					$T_s:=T_\mu^0:\mathcal{O}_\mu\rightarrow\mathcal{O}_{0}$ which are left and right adjoint to each other.
					Consider the endofunctor $\Theta_s=T_sT^s$ of $\mathcal{O}_0$. 
					Let $\mathscr{H}=\mathscr{H}(W,\mathfrak{h})$ be the Hecke category
					associated with $(W,S)$ and the realization $\mathfrak{h}$. Then one has a functor
					$\mathscr{H}\rightarrow \End(\Proj\mathcal{O}_0)$ sending $B_s$ to $\Theta_s$. This was shown 
					by Soergel \cite{Soe,Soe_HC} in terms of his bimodules, by embedding the subcategory of projectives 
					inside the category of modules over the coinvariant ring $C=R/R\cdot R^W_+$.
				\item\label{item_rmkpcan}
					Via the isomorphism of the theorem, the indecomposable objects correspond to a basis of $\mathbf{H}$ 
					over $\ZZ[v,v^{-1}]$. Consider realizations as in Example \ref{exa_realiz} \ref{item_exWf} and \ref{item_exW}.
					Then one can show (see for example \cite{JenWil}) that this basis only depends on
					the type of the root datum and on the characteristic $p$ of the residue field of 
					$\Bbbk$. This is the \emph{$p$-canonical} basis of $\mathbf{H}$ appearing in
 					character formulas for tilting and simple representations for the group $G$: 
 					see the work of Achar, Makisumi, Riche and 
 					Williamson \cite{AMRW_char} and of Riche and Williamson \cite{RW}.
 				\item\label{item_rmkSoergconj} Soergel \cite{Soe_HC} conjectured that, if $\Bbbk$ is a 
 					characteristic zero field, then the latter is the Kazhdan-Lusztig basis.
					For realizations over $\mathbb{R}$, this was proved by Elias and Williamson \cite{EWHodge}.
					This was usef to show, for a general Coxeter system, the positivity of the coefficients of 
					Kazhdan-Lusztig polynomials. Moreover Soergel had showed 
					that his conjecture implies Kazhdan-Lusztig's conjecture when $W$ is a Weyl group.
			\end{enumerate}
		\end{rmk}
		\section{Rouquier complexes}
		In this section we introduce Rouquier complexes in the homotopy category of the 
		Hecke category.
		\subsection{Definition}\label{subs_Roucom}
		Consider the homotopy category $\mathcal{K}^{\mathrm{b}}(\mathscr{H})$. 
		Recall that the monoidal structure of $\mathscr{H}$ extends to $\mathcal{K}^{\mathrm{b}}(\mathscr{H})$ as follows. 
		Let $A^\bullet,B^\bullet\in \mathcal{K}^{\mathrm{b}}(\mathscr{H})$, then:
		\[
			(A^\bullet\otimes B^\bullet)^i=\bigoplus_{p+q=i} A^p\otimes B^q,
		\]
		and the differential map, restricted to $A^p\otimes B^q$ is:
		\begin{equation}\label{eq_signconv}
			d\otimes \id + (-1)^p \id \otimes d.
		\end{equation}
		The unit is $\mathbbm{1}[0]$ (the object $\mathbbm{1}$ as a complex concentrated in degree 0).
			
		Consider the \emph{standard} and \emph{costandard} complexes $F_s$ and $F_s^{-1}$ 
		from the introduction.
		If $\sigma=\sigma_{s}^{\pm 1}\in \Sigma$, then let 
		$F_\sigma$ denote $F_s^{\pm 1}$.
		Then, for any braid word $\underline{\omega}=\sigma_1 \sigma_2 \dots \sigma_n$, with $\sigma_i\in \Sigma$,
		we put
		\begin{equation}\label{eq_defrouq}
			F_{\underline{\omega}}^\bullet:=F_{\sigma_1}\otimes F_{\sigma_2}\otimes \dots \otimes F_{\sigma_n}.		
		\end{equation}
		In the sequel we will often omit the symbol $\otimes$.
		These objects are called
	
		\emph{Rouquier complexes} and the subcategory $\mathscr{B}_W$ that they form is called \emph{2-braid group}. 
		\begin{rmk}
		These complexes were introduced in \cite{Rou_cat} in terms of Soergel bimodules 
		to study natural transformations between the endofunctors induced by braid group actions on categories. 
		As an example, in the setting of Remark \ref{rmk_onSoergelcat} \ref{item_catO}, 
		consider the complexes
		\begin{align*}
			\tilde{F}_s=\dots\rightarrow 0\rightarrow 0\rightarrow &\,\Theta_s\rightarrow \mathbbm{1}\rightarrow 0\rightarrow \dots\\
			\tilde{F}_s^{-1}=\dots\rightarrow 0\rightarrow\mathbbm{1} \rightarrow &\,\Theta_s\rightarrow 0\rightarrow 0\rightarrow\dots
		\end{align*}
		where the morphisms are
		given by the counit and unit of the two adjunctions between $T_s$ and $T^s$, 
		and where $\Theta_s$ is	in degree 0 for both. One can show that they are mutually inverse 
		endofunctors for the category $\mathcal{K}^{\mathrm{b}}(\Proj\mathcal{O}_0)=\mathcal{D}^{\mathrm{b}}(\mathcal{O}_0)$. 
		and the assignment $\sigma_s\mapsto\tilde{F}_s$ defines an action of $B_W$ on this category \cite[Theorem 10.4]{Rou_cat}. Furhtermore
		one can upgrade this to a functor from $\mathscr{B}_W$ to the category of self-equivalences
		of $\mathcal{D}^{\mathrm{b}}(\mathcal{O}_0)$ \cite[Theorem 10.5]{Rou_cat}. Hence morphisms between 
		Rouquier complexes give natural transformations between these endofunctors.
		\end{rmk}
		\begin{rmk}
		In the geometric setting (see Remark \ref{rmk_onSoergelcat} \ref{item_geom}), the 
		homotopy category $\mathcal{K}^{\mathrm{b}}(\mathscr{H})$ was considered by Achar and Riche 
		\cite[\S\,3.5]{AchRic_modII} as a modular version of the equivariant mixed derived category on 
		the (affine) flag variety. The standard and costandard complexes $F_s$ and $F_s^{-1}$ correspond to
		the standard and costandard sheaves $\Delta^{\mathrm{mix}}_{s}$ and $\nabla^{\mathrm{mix}}_{s}$ in that setting.
		\end{rmk}
		\subsection{Basic properties of Rouquier complexes}
		These properties were first proved by Rouquier \cite{Rou_cat} in the 
		language of Soergel bimodules. A sketch of proof is also given in \cite{Elias}. One can find a 
		diagrammatic argument in \cite{Mal_rou}.
		\begin{pro}\label{pro_braidRou}
			One has the following.
			\begin{enumerate}
			 	\item\label{item_propinvers} Let $s\in S$, then $F_s F_s^{-1}\cong F_s^{-1}F_s\cong \mathbbm{1}[0]$.
				\item\label{item_propbraid} Let $s,t\in S$ with $m_{st}<\infty$, then
					\[
						\underbrace{F_s F_tF_sF_t \cdots}_{m_{st} \text{ times}} \cong %
							\underbrace{F_t F_sF_tF_s \cdots}_{m_{st} \text{ times}}
					\]
			\end{enumerate}
			Hence, for each pair of braid words $\underline{\omega}_1$, $\underline{\omega}_2$ expressing
			the same element $\omega\in B_W$, there is an isomorphism $F_{\underline{\omega}_1}^\bullet\cong F_{\underline{\omega}_2}^\bullet$.
			Furthermore:
			\begin{enumerate}[resume]
				\item\label{item_propcanonic}(Rouquier Canonicity)
					for each $\underline{\omega}_1$ and $\underline{\omega}_2$ as above, we have 
					\[
						\Hom(F_{\underline{\omega}_1}^\bullet,F_{\underline{\omega}_2}^\bullet)\cong R,
					\]
					and one can chose
					$\gamma_{\underline{\omega}_1}^{\underline{\omega}_2}$ such that the system 
					$\{\gamma_{\underline{\omega}_1}^{\underline{\omega}_2}\}_{\underline{\omega}_1,\underline{\omega}_2}$ is transitive.
			\end{enumerate}
		\end{pro}
		\begin{rmk}\label{rmk_faithful}
			Thanks to these properties,	the Rouquier complexes $F_{\omega}$'s associated to $\omega\in B_W$ are 
			well defined up to a canonical isomorphism and they form a quotient
			of the braid group.	Rouquier conjectured that this quotient is just $B_W$ itself.
			This \emph{faithfulness} of the 2-braid group was shown in type $A$ by Khovanov and Seidel \cite{KhoSei}, in simply laced 
			finite type by Brav and Thomas \cite{BraTho}, and in all finite types by Jensen \cite{Jen}.
		\end{rmk}		
		We want to investigate representatives for $F_{\omega}$ in the category of complexes $\mathcal{C}^{\mathrm{b}}(\mathscr{H})$.
		First let us introduce some notation.
			\subsection{Notation for graded objects and complexes}
			Given a $\Bbbk$-linear additive category $\mathscr{C}$, a \emph{graded object} is a family $\{A^q\}_{q\in \mathbb{Z}}$,
			where $A^q$ is an object of $\mathscr{C}$ for all $q\in \mathbb{Z}$. If $A$ is an object of $\mathscr{C}$, then
			$A[-q]$ denotes the graded object whose only nonzero entry is $A$ in degree $q$.
			The direct sum of graded objects is defined in a natural way, so that we can also write
			\[
				\{A^q\}_{q\in\mathbb{Z}}=\bigoplus_{q\in \mathbb{Z}} A^q[-q].
			\] 
			A (degree 0) morphism of graded objects $A^\bullet=\{A^q\}$ and $B^\bullet=\{B^q\}$ is a family of morphisms
			\[
				\{f^q:A^q\rightarrow B^{q}\}_{q\in \mathbb{Z}}.
			\]
			This defines a category $\mathcal{G}_0(\mathscr{C})$
			of graded objects. The \emph{shift} functor $[1]$ sends the graded object $\{A^q\}_{q\in \mathbb{Z}}$ to
			$\{A^{q+1}\}_{q\in \mathbb{Z}}$. Then one can define the \emph{graded category} $\mathcal{G}(\mathscr{C})$ with the 
			same objects and morphism spaces
			\[
				\Hom_{\mathcal{G}(\mathscr{C})}(A^\bullet,B^\bullet):=\bigoplus_{p\in \mathbb{Z}} \Hom_{\mathcal{G}_0(\mathscr{C})}(A^\bullet,B^\bullet[p]).
			\]
			The $p$-th graded piece is denoted by $\Hom^p(A^\bullet,B^\bullet)$ and its elements are said 
			\emph{homogeneous} of degree $p$. 
			A complex in $\mathcal{C}(\mathscr{C})$ is then determined by a graded object $A^\bullet$ endowed with an
			endomorphism $d:A^\bullet\rightarrow A^\bullet$ of degree 1, such that $d^2=0$.
			Viceversa, let $\grad{(\cdot)}:\mathcal{C}(\mathscr{C})\rightarrow\mathcal{G}(\mathscr{C})$ denote 
			the forgetful functor sending $(A^\bullet,d)$ to $A^\bullet$.
			\subsection{Representatives for Rouquier complexes}\label{subs_heckcat_diagrams}
			The most immediate representatives for Rouquier complexes are just the tensor products \eqref{eq_defrouq}.
			Let us describe them more precisely.
			
			For a given subexpression $\mathbf{i}$ of a braid word $\underline{\omega}$, let $\underline{\omega}_{\mathbf{i}}$ denote the 
			subword corresponding to it. Then let $B_{\mathbf{i}}$ be the Bott-Samelson object 
			corresponding to the Coxeter projection of $\underline{\omega}_{\mathbf{i}}$. For example, given
			\[
				\underline{\omega}=\sigma_s\sigma_t\sigma_t\sigma_u^{-1},
			\]
			and $\mathbf{i}=0101$, we have 
			$\underline{\omega}_{\mathbf{i}}=\sigma_t\sigma_u^{-1}$ and then $B_{\mathbf{i}}=B_{\underline{tu}}=B_tB_u$.
			
			By definition of tensor product of complexes, we have
			\[
				\grad{(\Roudg{\underline{\omega}})}=\bigoplus_{\mathbf{i}\in\{0,1\}^{\ell(\underline{\omega})}}B_{\mathbf{i}}\Tate{q_{\mathbf{i}}},
			\]
			where $q_{\mathbf{i}}=\varpi(\underline{\omega})-\varpi(\underline{\omega}_{\mathbf{i}})$ and $\Tate{1}:=(1)[-1]$ denotes the \emph{Tate twist}.
			One can then compute the components of the differential map, according to
			 \eqref{eq_signconv}. The only nonzero components $B_{\mathbf{i}}\rightarrow B_{\mathbf{i}'}$ are those 
			 for which $\mathbf{i}'$ is obtained from $\mathbf{i}$ by changing one symbol in such a way that 
			 $q_{\mathbf{i}'}=q_{\mathbf{i}}+1$.
			 In this case the component is either
			 \begin{align*}
			 	&(-1)^k	 	
					\begin{tikzpicture}[x=0.3cm,y=0.4cm,baseline=0.36cm,every node/.style={font=\scriptsize}]
						\draw[gray] (0.5,0)--(9.5,0);\draw[gray] (0.5,2)--(9.5,2);
						\foreach \i in {1,2,4,6,8,9}{%
								\draw[violet] (\i,0)--(\i,2);
							}
						\pic[violet] at (5,0) {dot={1cm}};
						\node[font=\normalsize] at (3,1) {\dots}; \node[font=\normalsize] at (7,1) {\dots};
					\end{tikzpicture}&
					&\text{or}&
					&(-1)^k
						\begin{tikzpicture}[x=0.3cm,y=0.4cm,baseline=0.36cm,every node/.style={font=\scriptsize}]
							\draw[gray] (0.5,0)--(9.5,0);\draw[gray] (0.5,2)--(9.5,2);
							\foreach \i in {1,2,4,6,8,9}{%
									\draw[violet] (\i,0)--(\i,2);
								}
							\pic[violet,yscale=-1] at (5,2) {dot={1cm}};
							\node[font=\normalsize] at (3,1) {\dots}; \node[font=\normalsize] at (7,1) {\dots};
						\end{tikzpicture}.&
			 \end{align*}
			 where $k$ is the number of 0's in $\mathbf{i}$ preceding the changed symbol.

			Then the complex $F_{\underline{\omega}}^\bullet$ has the form 
			of a cube of dimension equal the number of letters in $\underline{\omega}$: the vertices correspond to the 
			Bott-Samelson objects obtained from all possible subexpressions of $\underline{\omega}$ and the edges are the arrows 
			describing the component of the differential map.
			\begin{exa}\label{exa_cube}
				Let $\re{s},\bl{t}\in S$ and 
				$\underline{\omega}=\sigma_s\sigma_s\sigma_t^{-1}$. Then the complex $\Roudg{\underline{\omega}}=F_sF_sF_t^{-1}$ 
				is the following:
				\begin{center}
					\begin{tikzpicture}[x=2.7cm,y=0.9cm,every pic/.style={x=1cm,scale=0.5,font=\tiny,-}]
						\summandnode{ss9}{0}{0}{ss9};
						\summandnode{sst}{1}{2}{sst};\summandnode{s}{1}{0}{s1};\summandnode{s}{1}{-2}{s2};
						\node (s1t1) at (2,2) {$B_sB_t(1)$};
						\node (s2t1) at (2,0) {$B_sB_t(1)$};
						\node (n1) at (2,-2) {$\mathbbm{1}(1)$};
						\summandnode{t2}{3}{0}{t2};
						\draw[-latex] (ss9) to %
							pic[midway,yshift=.5cm]{bottomdot={sst}{3}{+}} (sst);%
						\draw[-latex] (ss9) to%
							pic[midway,yshift=.3cm]{topdot={ss}{2}{+}} (s1);%
						\draw[-latex] (ss9) to%
							pic[midway,yshift=-.5cm]{topdot={ss}{1}{+}} (s2);%
						\draw[-latex] (sst) to%
							pic[midway,yshift=.3cm]{topdot={sst}{2}{+}} (s1t1);%
						\draw[-latex] (sst) to%
							pic[pos=.6,xshift=.8cm]{topdot={sst}{1}{+}} (s2t1);%
						\draw[-latex] (s1) to%
							pic[pos=.4,xshift=-.6cm]{bottomdot={st}{2}{-}} (s1t1);%
						\draw[-latex] (s1) to%
							pic[near start,xshift=-.3cm]{topdot={s}{1}{+}} (n1);%
						\draw[-latex] (s2) to%
							pic[pos=.6,xshift=.8cm]{bottomdot={st}{2}{-}} (s2t1);%
						\draw[-latex] (s2) to%
							pic[midway,yshift=-.3cm]{topdot={s}{1}{-}} (n1);%
						\draw[-latex] (s1t1) to%
							pic[midway,yshift=.5cm]{topdot={st}{1}{+}} (t2);%
						\draw[-latex] (s2t1) to%
							pic[midway,yshift=.3cm]{topdot={st}{1}{-}} (t2);%
						\draw[-latex] (n1) to%
							pic[midway,yshift=-.5cm]{bottomdot={t}{1}{+}} (t2);%
						\node[green,font=\small] at (0,-3){-1}; \node[green,font=\small] at (1,-3){0};%
						\node[green,font=\small] at (2,-3){1}; \node[green,font=\small] at (3,-3) {2};
						\node at (1,1) {$\oplus$}; \node at (1,-1) {$\oplus$};
						\node at (2,1) {$\oplus$}; \node at (2,-1) {$\oplus$};
					\end{tikzpicture}		
				\end{center}			
			\end{exa}	  			 			
			When the category $\mathscr{H}$ is Krull-Schmidt, one can 
			get rid of a maximal null-homotopic summand of each $F_{\underline{\omega}}^\bullet$ and obtain
			its \emph{minimal subcomplex} representing $F_{\omega}$. 
			This is a summand which does not admit null-homotopic factor and one can show that it is
			unique up to isomorphism (and not just homotopy equivalence): see \cite[\S\,6.1]{EWHodge}.

			Minimal subcomplexes of Rouquier complexes are hard to find in general and for arbitrary 
			coefficients, this notion is not well defined. 
			For $\underline{\omega}$ positive (or negative) we will now describe a representative $F_{\underline{\omega}}$
			for $F_{\omega}$ which is simpler than $F_{\underline{\omega}}^\bullet$, and available 
			without restrictions on $\Bbbk$. This can be consider an intermediate step towards a minimal subcomplex.
			This is obtained by getting rid of a large 
			null-homotopic summand, via Gaussian elimination for complexes.
			The graded object of the new complex is a direct sum over subwords, rather than subexpressions.
			\section{Reduced representatives for Rouquier complexes}
			Let us start with a simple example to illustrate our result.
			\begin{exa}\label{exa_sss}
				Let $\underline{\omega}=\sigma_{\re{s}}\sigma_{\re{s}}\sigma_{\re{s}}$. Then $F_{\underline{\omega}}^\bullet$ is
				\[
					\begin{tikzcd}[row sep=tiny]
																	& B_{\underline{ss}}(1) \ar[r]\ar[rdd]		& B_s(2)\ar[rdd]	& \\
																	& \oplus									& \oplus			& \\
						B_{\underline{sss}} \ar[ruu]\ar[r]\ar[rdd]	& B_{\underline{ss}}(1) \ar[ruu]\ar[rdd]		& B_s(2)\ar[r]	& \mathbbm{1}(3)\\
																	& \oplus									& \oplus			& \\
																	& B_{\underline{ss}}(1) \ar[ruu]\ar[r]		& B_s(2)\ar[ruu]	& \\
					\end{tikzcd}
				\]
				According to Example \ref{exa_basicdeco}, the Bott-Samelson objects appearing above decompose as
				\begin{align*}
					&B_{\underline{ss}}(1)=B_s\oplus B_s(2), &			
					&B_{\underline{sss}}=B_s(-2)\oplus B_s^{\oplus 2} \oplus B_s(2).
				\end{align*}
				Hence the complex can be written
				\begin{equation}\label{eq_exadecomcol}
						\begin{tikzcd}[row sep=tiny]
																												& \bl{B_s}\oplus \gr{B_s(2)} \ar[r]\ar[rdd]		& \gr{B_s(2)}\ar[rdd]	& \\
																												& \oplus										& \oplus				& \\
							\textcolor{violet}{B_s(-2)}\oplus \bl{B_s^{\oplus 2}} \oplus \gr{B_s(2)} \ar[ruu]\ar[r]\ar[rdd]	& \bl{B_s}\oplus \gr{B_s(2)} \ar[ruu]\ar[rdd]	& \gr{B_s(2)}\ar[r]		& \mathbbm{1}(3)\\
																												& \oplus										& \oplus				& \\
																												& \bl{B_s}\oplus \gr{B_s(2)} \ar[ruu]\ar[r]		& \gr{B_s(2)}\ar[ruu]	& \\
						\end{tikzcd}
				\end{equation}
				and can be filtered by the subquotients:
				\begin{align*}
					&\textcolor{violet}{G_3},& &\bl{G_2},& &\gr{G_1},& &G_0,&
				\end{align*}
				corresponding to the colors in the above picture (for a precise definition of subquotient in this setting, 
				see below).
				One can now check that the components of the differential map in each $G_i$ are just identities. For instance, the blue and
				the green pieces are
				\begin{align*}
					&\begin{tikzcd}[ampersand replacement=\&,row sep=tiny]
												\& \bl{B_s} \\
						\bl{B_s}\ar[ru]\ar[rd]	\& \oplus	\\
						\oplus					\& \bl{B_s} \\
						\bl{B_s}\ar[ru]\ar[rd]	\& \oplus \\
												\& \bl{B_s}
					\end{tikzcd}&
					&\begin{tikzcd}[ampersand replacement=\&,row sep=tiny]
															\& \gr{B_s(2)}\ar[r]\ar[rdd]		\& \gr{B_s(2)} \\
															\& \oplus						\& \oplus		\\
						\gr{B_s(2)}\ar[ruu]\ar[r]\ar[rdd]	\& \gr{B_s(2)}\ar[ruu]\ar[rdd]	\& \gr{B_s(2)} \\
															\& \oplus						\& \oplus	 \\
															\& \gr{B_s(2)}\ar[ruu]\ar[r]		\& \gr{B_s(2)}\\
					\end{tikzcd}
				\end{align*}
				Notice that we can describe these complexes by simplicial sets whose faces correspond to some
				$B_s(i)$, and where face relationship corresponds to the arrows. More precisely, 
				consider the following simplicial sets.
				\begin{align*}
					& \begin{tikzpicture}
						\draw[blue] (1,0)--(2,0)--(3,0);
						\foreach \i in {1,2,3} {%
							\fill[blue] (\i,0) circle (1.5pt);
							\draw (\i,0) circle (1.5pt); 
						}
					\end{tikzpicture}&
					& \begin{tikzpicture}[baseline=0]
						\fill[green,opacity=.4] (-30:1)--(90:1)--(210:1)--cycle;
						\draw[green] (-30:1)--(90:1)--(210:1)--cycle;
						\foreach \i in {-30,90,210} {%
							\fill[green] (\i:1) circle (1.5pt);
							\draw (\i:1) circle (1.5pt); 
						}
					\end{tikzpicture}&
				\end{align*}
				Namely, in the blue picture, the edges correspond to the two $B_s$'s in lower cohomological degree and the 
				three vertices to the others, and similarly in the green picture.
				
				One can then use Gaussian elimination (see below) to reduce these complexes: roughly speaking, 
				one can eliminate arrows which are isomorphisms and obtain a summand of the original complex
				which is homotopy equivalent to it. This corresponds, in the simplicial sets, to an \emph{elementary collapse}:
				the removal of a simplex together with one of its faces (see the precise definition below). 
				For example, for the green part, we can perform three
				of these collapses, as shown in the following picture (the removed parts are drawn in gray):
				\[
					\begin{tikzpicture}[baseline=0]
						\fill[green,opacity=.4] (-30:1)--(90:1)--(210:1)--cycle;
						\draw[green] (-30:1)--(90:1)--(210:1)--cycle;
						\foreach \i in {-30,90,210} {%
							\fill[green] (\i:1) circle (1.5pt);
							\draw (\i:1) circle (1.5pt); 
						}
					\end{tikzpicture}
					\quad\rightsquigarrow\quad
					\begin{tikzpicture}[baseline=0]
						\fill[gray,opacity=.1] (-30:1)--(90:1)--(210:1)--cycle;
						\draw[green] (-30:1)--(90:1)--(210:1);
						\draw[dashed,gray] (-30:1)--(210:1);
						\foreach \i in {-30,90,210} {%
							\fill[green] (\i:1) circle (1.5pt);
							\draw (\i:1) circle (1.5pt); 
						}
					\end{tikzpicture}
					\quad\rightsquigarrow\quad
					\begin{tikzpicture}[baseline=0]
						\fill[gray,opacity=.1] (-30:1)--(90:1)--(210:1)--cycle;
						\draw[green] (-30:1)--(90:1);
						\draw[gray,dashed] (90:1)--(210:1)--(-30:1);
						\foreach \i in {-30,90} {%
							\fill[green] (\i:1) circle (1.5pt);
							\draw (\i:1) circle (1.5pt); 
						}
						\fill[white] (210:1) circle (1.5pt);
						\draw[gray] (210:1) circle (1.5pt); 
					\end{tikzpicture}
					\quad\rightsquigarrow\quad
					\begin{tikzpicture}[baseline=0]
						\fill[gray,opacity=.1] (-30:1)--(90:1)--(210:1)--cycle;
						\draw[gray,dashed] (90:1)--(210:1)--(-30:1)--cycle;
						\foreach \i in {-30} {%
							\fill[green] (\i:1) circle (1.5pt);
							\draw (\i:1) circle (1.5pt); 
						}
						\foreach \i in {90,210}{
							\fill[white] (\i:1) circle (1.5pt);
							\draw[gray] (\i:1) circle (1.5pt); 
						}
					\end{tikzpicture}					
				\]
				This transforms the green complex into $B_s(2)[-2]$.
				One can then show that all these eliminations can be performed together: each simplicial set can be collapsed to a point,
				and the original complex can be reduced to
				\[
					\begin{tikzcd}
						\textcolor{violet}{B_s(-2)}\ar[r] & \bl{B_s}\ar[r] & \gr{B_s(2)}\ar[r] & \mathbbm{1}(3)
					\end{tikzcd}
				\]
				Furthermore the procedure gives formulas for the projection corresponding to this summand
				and allows to compute the new differential map. In this case this is given by
				the following three morphisms:
				\begin{align*}
					& \delta_s\, \begin{tikzpicture}[baseline=-0.1cm,y=.5cm]
									\draw[red](0,-1)--(0,1);
								\end{tikzpicture}
						\hphantom{\delta_s}
						-
						\hphantom{\delta_s}
						\begin{tikzpicture}[baseline=-0.1cm,y=.5cm]
									\draw[red](0,-1)--(0,1);
								\end{tikzpicture}
								\,s(\delta_s)&
					& \delta_s\, \begin{tikzpicture}[baseline=-0.1cm,y=.5cm]
									\draw[red](0,-1)--(0,1);
								\end{tikzpicture}
						\hphantom{\delta_s}
						-
						\hphantom{\delta_s}
						\begin{tikzpicture}[baseline=-0.1cm,y=.5cm]
									\draw[red](0,-1)--(0,1);
						\end{tikzpicture}
						\,\delta_s&
					& \begin{tikzpicture}[baseline=-0.1cm,y=.5cm]
									\draw[red](0,-1)--(0,0);
									\fill[red] (0,0) circle (1.5pt);
								\end{tikzpicture}
				\end{align*}
				Finally, notice that each of the colored part that we defined above corresponds to 
				a subword of $sss$, and reduces to a single object in the final complex.
			\end{exa}
			We would now like to make this procedure more precise and generalize it to any positive 
			Rouquier complex. 
			\subsection{Large scale Gaussian elimination for complexes}\label{subs_gausselim}
			Our main tool is Gaussian elimination for 
			complexes. Let $\mathscr{C}$ be an additive category.			
			A summand of a complex in the category 
			$\mathcal{C}^{\mathrm{b}}(\mathscr{C})$ will be called \emph{Gaussian} if the projection and the inclusion morphisms
			are mutually inverse homotopy equivalences.
			 
			The following was first pointed out by Bar-Natan \cite{BarNat}.
			\begin{lem}\label{lem_Gausselimination}
				Consider the following complex in $\mathcal{C}^{\mathrm{b}}(\mathscr{C})$ 
				\[
					\begin{tikzcd}[ampersand replacement=\&,column sep=large,baseline=.15cm]
						A \arrow{r}{ \begin{psmallmatrix}
										\epsilon \\ \zeta
				 					 \end{psmallmatrix} }		\& B\oplus E \arrow{r}{ \begin{psmallmatrix}
																						\alpha	& \beta \\
																						\gamma	& \phi 
																					\end{psmallmatrix} } \& C\oplus E' \arrow{r}{ 	\begin{psmallmatrix}
																																		\eta	& \theta \\
																																	\end{psmallmatrix} } 	\& D
					\end{tikzcd}
				\]
				and suppose that $\phi: E\rightarrow E'$ is an isomorphism in $\mathscr{C}$. 
				Then the following 
				\[
					\begin{tikzcd}[column sep=large]
					 A\ar[r,"\epsilon"] & B\ar[r,"\alpha-\beta\phi^{-1}\gamma"] & C\ar[r,"\eta"] & D
					\end{tikzcd}
				\]
				is a Gaussian summand.
				\end{lem}
			\begin{proof}
				Consider the projection morphism $\pi$ given by:
				\begin{equation}\label{eq_projmorph}
					\begin{tikzcd}[ampersand replacement=\&,column sep=large,baseline=.15cm]
						A \arrow{r}{ \begin{psmallmatrix}
										\epsilon \\ \zeta
				 					 \end{psmallmatrix} }\ar[d,equal]		\& B\oplus E \arrow{r}{ \begin{psmallmatrix}
																						\alpha	& \beta \\
																						\gamma	& \phi 
																					\end{psmallmatrix} }
																						\arrow{d}{\begin{psmallmatrix}
																									\id_B & 0
																								\end{psmallmatrix}}
																												\& C\oplus E' \arrow{r}{ 	\begin{psmallmatrix}
																																		\eta	& \theta \\
																																	\end{psmallmatrix} }
																																\arrow{d}{\begin{psmallmatrix}
																																	\id_C & -\beta\phi^{-1}
																																\end{psmallmatrix}} 	\& D\ar[d,equal]\\
						A \ar[r,swap,"\epsilon"]		\& B \ar[r,swap,"\alpha-\beta\phi^{-1} \gamma"]	\& C \ar[r,swap,"\eta"]	\& D
					\end{tikzcd}			
				\end{equation}
				and the inclusion morphism $\iota$ described by:
				\[
					\begin{tikzcd}[ampersand replacement=\&,column sep=large,baseline=.15cm]
						A \arrow{r}{ \begin{psmallmatrix}
										\epsilon \\ \zeta
				 					 \end{psmallmatrix} }		\& B\oplus E \arrow{r}{ \begin{psmallmatrix}
																						\alpha	& \beta \\
																						\gamma	& \phi 
																					\end{psmallmatrix} } \& C\oplus E' \arrow{r}{ 	\begin{psmallmatrix}
																																		\eta	& \theta \\
																																	\end{psmallmatrix} }\& D \\
						A \ar[r,swap,"\epsilon"]\ar[u,equal]		\& B \ar[r,swap,"\alpha-\beta\phi^{-1} \gamma"]	\arrow{u}{\begin{psmallmatrix}
																																\id_B \\ -\phi^{-1}\gamma
																															\end{psmallmatrix}}	\& C \ar[r,swap,"\eta"]	\arrow{u}{\begin{psmallmatrix}
																																											\id_C \\ 0
																																										\end{psmallmatrix}} 	\& D \ar[u,equal]
					\end{tikzcd}			
				\]
				It is easy to check that $\pi\iota=\id$. The 
				idempotent $\iota\pi$ is homotopy equivalent to the identity because
				\[
					\begin{tikzcd}[ampersand replacement=\&,row sep=huge,column sep=huge]
						A \ar[r] \& B\oplus E \ar[r]\ar[ld,swap,"0"] \& C\oplus E' \ar[r]\arrow{ld}[swap,pos=0.6]{\begin{psmallmatrix}
																												0 & 0 \\
																												0 & \phi^{-1}
																											\end{psmallmatrix}}		\& D \ar[ld,swap,"0"] \\				
						A \ar[r] \& B\oplus E \ar[r] 			\& C\oplus E' \ar[r]												\& D 
					\end{tikzcd}
				\]				
				is a homotopy from the complementary idempotent to the zero morphism.
			\end{proof}
			We want now to describe a way to perform many Gaussian eliminations on complexes where the
			isomorphisms that fit together in a nice way, as in in Example \ref{exa_sss}.
			
			Recall that a \emph{polytopal set}\footnote{The standard terminology (see for example \cite[\S\,8.1]{Zieg})
			is \emph{polytopal complex} but in our context the word ``complex'' would be
			overloaded.} of dimension $n$ is a finite collection $\Pi$ of convex polytopes in 
			$\mathbb{R}^d$ of dimensions up to $n$ such that:
			\begin{enumerate}
				\item a face of a polytope $\sigma\in \Pi$ belongs to $\Pi$;
				\item if $\sigma,\sigma'\in \Pi$ then the intersection $\sigma\cap\sigma'$ is either
					empty or a common face of $\sigma$ and $\sigma'$ (so in particular it belongs to $\Pi$).
			\end{enumerate}	 
			Let $\Pi^i$ denote the subset of $i$-dimensional polytopes of $\Pi$.
			Here 0-dimensional polytopes are just points.
			This generalizes the notion of simplicial complex.
			
			We can now make the above idea more precise.
			\begin{dfn}
				Let $\Pi$ be a polytopal set of dimension $n$, and let $A$ be a complex in 
				$\mathcal{C}^{\mathrm{b}}(\mathscr{C})$. We say that $A$ is \emph{left-described} by $\Pi$ if:
				\begin{enumerate}
					\item as a graded object, the complex $A$ decomposes as
						\[
							\bigoplus_{\sigma\in \Pi} A_\sigma[\dim \sigma]
						\]
						with $A_\sigma$ in $\mathscr{C}$.
						So lower dimensional polytopes correspond to objects in higher cohomological degrees.
					\item the component $\phi_{\tau,\sigma}:A_{\sigma}\rightarrow A_{\tau}$ of the 
						differential map is non-zero if and only if $\tau$ is a face of $\sigma$.
					\item all non-zero components of the differential map are isomorphisms.	
				\end{enumerate}
			\end{dfn}	
			One can think of the objects $A_\sigma$ as ``living over'' the polytopes of $\Pi$.
			The dual notion of \emph{right-description} can be obtained by reversing all the arrows.
			The face relationship is also reversed: in a right-description higher dimensional polytopes correspond 
			to objects in higher cohomological degrees.
			
			Consider the 1-skeleton $\Pi^0\cup \Pi^1$ of a polytopal set $\Pi$ which left-describes
			a complex $A$. If $e$ is an oriented edge 
			from the vertex $p$ to the vertex $q$, then we have isomorphisms $\phi_{p,e}$ and $\phi_{q,e}$.
			We define the \textit{isomorphism associated to the oriented edge $e$} to be
			\[
				\Phi(e) = - \phi_{q,e}(\phi_{p,e})^{-1}.
			\]
			We put a minus sign for a technical reason that will become clear later.
			Notice that if $-e$ is the same edge with opposite orientation, then 
			$\Phi(-e)=\Phi(e)^{-1}$.
			Given an oriented path $P$ through the oriented edges $e_1,\dots e_r$ along the 
			1-skeleton of $\Pi$,
			the \textit{isomorphism associated to $P$} is the composition 
			\[
				\Phi(P):=\Phi(e_r) \cdots \Phi(e_1).
			\]
			Clearly, if $P'$ is another path starting from the endpoint of $P$, and
			$P''$ is the concatenation of $P$ and $P'$, then
			\[
				\Phi(P'')=\Phi(P')\Phi(P).
			\]
			\begin{lem}\label{lem_homotopypath}
				If two paths $P$ and $P'$ connecting the point $q$ to $p$ are homotopic in $\Pi$ 
				then $\Phi(P)=\Phi(P')$.
			\end{lem}
			\begin{proof}
			It is sufficient to 
			prove that the isomorphism associated to a contractible loop $P$
		
			is the identity. Such a loop is the 
			boundary of the union of a finite number of polygons.
			By splitting the isomorphism as shown below, 
			\[
				\begin{tikzpicture}[baseline=1cm,every node/.style={inner sep=.1cm}]
					\coordinate (p0) at (0,0); \node[anchor=north east] (pp0) at (p0) {};
					\coordinate (p1) at (-1,1); \node[anchor=north east] (pp1) at (p1) {};
					\coordinate (p2) at (-1,2); \node[anchor=south east] (pp2) at (p2) {};
					\coordinate (p3) at (0,3); \node[anchor=south east] (pp3) at (p3) {};\node[anchor=south west] (pp31) at (p3) {};
					\coordinate (p4) at (2,2); \node[anchor=south west] (pp4) at (p4) {};
					\coordinate (p5) at (3,1); \node[anchor=south west] (pp5) at (p5) {};\node[anchor=north west] (pp51) at (p5) {};
					\coordinate (p6) at (2,0); \node[anchor=north west] (pp6) at (p6) {};
					\fill[green!40](p0)--(p1)--(p2)--(p3)--(p4)--(p5)--(p6)--cycle;
					\draw[green](p0)--(p1)--(p2)--(p3)--(p4)--(p5)--(p6)--cycle;
					\draw[green] (p0)--(p3); \draw[green] (p0)--(p4);
					\draw[->,thick] (pp0)--(pp1);	
					\draw[->,thick] (pp1)--(pp2); 
					\draw[->,thick] (pp2)--(pp3);
					\draw[->,thick] (pp31)--(pp4); 
					\draw[->,thick] (pp4)--(pp5); 
					\draw[->,thick] (pp51)--(pp6);
					\draw[->,thick] (pp6)--(pp0);
				\end{tikzpicture}
				\quad\rightsquigarrow\quad
				\begin{tikzpicture}[baseline=1cm,every node/.style={inner sep=.1cm}]
					\coordinate (p01) at (0,0); \node[anchor=north east] (pp01) at (p01) {};\node[anchor=north west] (pp011) at (p01) {};
					\coordinate (p11) at (-1,1);\node[anchor=north east] (pp11) at (p11) {};
					\coordinate (p21) at (-1,2); \node[anchor=south east] (pp21) at (p21) {};
					\coordinate (p31) at (0,3); \node[anchor=south east] (pp31) at (p31) {};\node[anchor=south west] (pp311) at (p31) {};
					\coordinate (p02) at (.5,0.25); \node[anchor=north east] (pp02) at (p02) {};\node[anchor=north west] (pp02bis) at (p02) {};
					\coordinate (p32) at (.5,3.25); \node[anchor=south east] (pp32) at (p32) {};\node[anchor=south west] (pp32bis) at (p32) {};
					\coordinate (p42) at (2.5,2.25); \node[anchor=south west] (pp42) at (p42) {}; \node[anchor=north west] (pp42bis) at (p42) {};
					\coordinate (p03) at (1,0); \node[anchor=south east] (pp03) at (p03) {};\node[anchor=north east] (pp03bis) at (p03) {};
					\coordinate (p43) at (3,2); \node[anchor=south east] (pp43) at (p43) {};\node[anchor=south west] (pp43bis) at (p43) {};
					\coordinate (p53) at (4,1); \node[anchor=south west] (pp53) at (p53) {};\node[anchor=north west] (pp53bis) at (p53) {};
					\coordinate (p63) at (3,0); \node[anchor=north west] (pp63) at (p63) {};
					\fill[green!40](p01)--(p11)--(p21)--(p31)--cycle;
					\fill[green!40](p02)--(p32)--(p42)--cycle;
					\fill[green!40](p03)--(p43)--(p53)--(p63)--cycle;
					\draw[green](p01)--(p11)--(p21)--(p31)--cycle;
					\draw[green](p02)--(p32)--(p42)--cycle;
					\draw[green](p03)--(p43)--(p53)--(p63)--cycle;
					\draw[->,thick] (pp01)--(pp11);
					\draw[->,thick] (pp11)--(pp21);
					\draw[->,thick] (pp21)--(pp31);
					\draw[->,thick,red] (pp311)--(pp011);
					\draw[->,thick,red] (pp02)--(pp32);
					\draw[->,thick] (pp32bis)--(pp42);
					\draw[->,thick,red] (pp42bis)--(pp02bis);
					\draw[->,thick,red] (pp03)--(pp43);
					\draw[->,thick] (pp43bis)--(pp53);
					\draw[->,thick] (pp53bis)--(pp63);
					\draw[->,thick] (pp63)--(pp03bis);
				\end{tikzpicture}
			\]
			we can restrict to the case where $P$ is the boundary of a single polygon $\sigma$.
			Let $e_1,\dots,e_r$ be the consecutive oriented edges forming $P$, and let
			$q_0,\dots, q_{r-1},q_r=q_0$ be its vertices, so that $e_i$ goes from $q_{i-1}$ to
			$q_i$. Then
			\begin{equation}\label{eq_pathpolygon}
				\Phi(P)=\Phi(e_r)\dots\Phi(e_1)=(-1)^r \phi_{q_r,e_r}\phi_{q_{r-1},e_r}^{-1}\dots\phi_{q_1,e_1}\phi_{q_0,e_1}^{-1}.			
			\end{equation}
			The square of the differential map vanishes, so, computing the component
			$A_{\sigma}\rightarrow A_{q_i}$ of $d^2$, we have
			\[
				\phi_{q_i,e_{i+1}}\phi_{e_{i+1},\sigma}+\phi_{q_{i},e_i}\phi_{e_i,\sigma}=0.
			\]
			This implies
			\begin{equation}\label{eq_replacement}
				\phi_{q_i,e_{i+1}}^{-1}\phi_{q_{i},e_i}=-\phi_{e_{i+1},\sigma}\phi_{e_i,\sigma}^{-1}			
			\end{equation}
			and, applying \eqref{eq_replacement} to \eqref{eq_pathpolygon}, we get the identity.			
			\end{proof}
			A face of a polytope of $\Pi$ is called \emph{free} if it is not the face of 
			any other polytope of $\Pi$. A \emph{collapse} is the operation of removing 
			from $\Pi$ a polytope $\sigma$ of maximal dimension together with a free face $\tau$ and all the faces of $\sigma$ containing $\tau$.
			A collapse is called \emph{elementary} if the face $\tau$ has codimension 1 (so we are only removing $\sigma$ and $\tau$). One can show
			that any collapse can be obtained by a sequence of elementary collapses.
			Then $\Pi$ is called \emph{collapsible} if there is a sequence of (elementary) collapses that reduces it to a single point.
			
			We can now give the key lemma for large scale Gaussian elimination.
			\begin{lem}\label{corContrac}
				Let $A$ be a complex left-described by a polytopal set $\Pi$.
				If $\Pi$ is collapsible to the point $p$, then $A_p[0]$ is a Gaussian summand of $A$.
				Furthermore, if one chooses a path $P_q$ from each point $q$ to $p$, then 
				the projection morphism $\pi$ is given componentwise by
				\[
					\begin{cases}
						\pi_{q}=\Phi(P_q) & \text{if $q\in \Pi^0$,} \\
						\pi_{\sigma} = 0 & \text{if $\sigma\notin \Pi^0.$}
					\end{cases}
				\]
			\end{lem}
			\begin{proof}
			Notice that the elementary collapse of a face $\sigma$ with a free face $\tau$ in $\Pi$
			corresponds to the elimination of the isomorphism $\phi_{\tau,\sigma}$ in $A$. 
			The condition on $\tau$ to be free corresponds to the condition $\phi_{\sigma',\tau}=0$ 
			for any $\sigma'\neq \sigma$.
			In other words, the complex is of the form
			\[
				\begin{tikzcd}[ampersand replacement=\&]
					\dots \ar[r] \&
					\Big(\bigoplus_{\sigma'\in\Pi^i\setminus\{\sigma\}}A_{\sigma'}\Big)\oplus A_{\sigma} 
						\arrow{r}{\begin{psmallmatrix}
									* & * \\ 0 & \phi_{\tau,\sigma}
								\end{psmallmatrix}} \&
						\Big(\bigoplus_{\tau'\in\Pi^{i-1}\setminus\{\tau\}}A_{\tau'}\Big)\oplus A_{\tau} 
						\ar[r] \&\dots				
				\end{tikzcd}
			\]
			Hence, by Lemma \ref{lem_Gausselimination}, we can eliminate $A_\sigma$ and $A_\tau$ and  
			the projection, according to \eqref{eq_projmorph}, behaves as follows:
			\begin{itemize}
				\item it is the identity on all $A_{\sigma'}$ for $\sigma'\neq\sigma,\tau$;
				\item it is zero on $A_\sigma$;
				\item it sends $A_{\tau}$ on each $A_{\tau'}$ via 
					$-\phi_{\tau',\sigma}\phi_{\tau,\sigma}^{-1}$.
			\end{itemize}
			The Gaussian summand is now the complex left-described by $\Pi\setminus \{\sigma,\tau\}$. 
			Then we can proceed by induction.
						
			If we compose all the projections thus obtained, we see that all $A_\sigma$ with $\sigma\notin \Pi^0$ are
			eventually sent to zero, whereas the $A_q$ for $q\in \Pi^0$ are sent to $A_p$ by a composition of
			$-\phi_{q',e}\phi_{q,e}^{-1}$ which gives precisely a path from $q$ to $p$.
			
			Any two paths connecting $q$ to $p$ are homotopic by contractibility (collapsible polytopal sets are clearly contractible).
			Hence the morphism $\Phi(P_q)$ does not depend on the choice of the path, by Lemma \ref{lem_homotopypath}. 
			\end{proof}
			Of course one can reverse all the arrows in the lemma, and consider inclusion morphism instead of 
			projection, and obtain an analogous statement for 
			right-described complexes.
			\subsection{Exact structure for complexes}
			We want now to give a precise meaning to filtrations in the category $\mathcal{C}^{\mathrm{b}}(\mathscr{C})$, following \cite[\S\,2-4]{Verd}.
			\begin{dfn}
				We say that a sequence $A\rightarrow B \rightarrow C$
				is \emph{exact} in $\mathcal{C}^{\mathrm{b}}(\mathscr{C})$ if it is split exact in the additive category $\mathcal{G}(\mathscr{C})$.
				In this case we say that $A$ is a \emph{subcomplex} of $B$ and $C$ is a \emph{quotient}.
			\end{dfn}
			This defines an exact structure on $\mathcal{C}^{\mathrm{b}}(\mathscr{C})$ in the sense of \cite{Quil}. Notice that when we pass to
			the homotopy category, exact sequences become distinguished triangles.
			
			We say that complexes $A_i$ ($i=0,\dots,n$) \emph{filter} $A$ by subcomplexes when 
			$A_i$ is a subcomplex of $A_{i+1}$, $A_0=0$ and $A_n=A$. Then the quotients
			of $A_{i+1}$ by $A_i$ are called \emph{subquotients} of the filtration.
			\begin{lem}
				If $A$ and $B$ are complexes filtered by $A_i$ and $B_i$ respectively, 
				and $f:A\rightarrow B$ a morphism compatible with the filtrations. Then $f$ is
				the projection to (inclusion of) a Gaussian summand if and only if it is so on all subquotients.
			\end{lem}
			\begin{proof}
				It suffices to prove that, given a commutative diagram
				\[
					\begin{tikzcd}
						X\ar[r]\ar[d]	& Y \ar[r]\ar[d]	& Z \ar[d] \\
						X'\ar[r]		& Y' \ar[r]			& Z'
					\end{tikzcd}
				\]
				where the rows are exact, if two of the vertical arrows are projections to a Gaussian summand, then
				the third is too. 
				Complete the diagram to a square
				\[
					\begin{tikzcd}
						X''\ar[r]\ar[d]	& Y''\ar[r]\ar[d]	& Z''\ar[d]\\
						X\ar[r]\ar[d]	& Y\ar[r]\ar[d]		& Z \ar[d]\\
						X'\ar[r]		& Y'\ar[r]			& Z'
					\end{tikzcd}
				\]
				where all rows and columns are exact (this is possible by the exact category versions of
				the $3\times 3$ lemma and the snake lemma). Now the first and last columns are split by hypothesis and 
				$X''$ and $Z''$ are null-homotopic, because $X'$ and $Z'$ are Gaussian. Then also $Y''$ is null-homotopic
				(it suffices to see the top row as a distinguished triangle in the homotopy category). 
				But then the middle	column has to be split too (contractible complexes are injective/projective
				for the Frobenius structure on $\mathcal{C}^{\mathrm{b}}(\mathscr{C})$: see for example \cite{Kell_Der}). 
				The case of inclusions is symmetric.
			\end{proof}
			\subsection{Reduced representatives}\label{subs_reducedcom}
			Let us now describe the reduced version for positive Rouquier complexes more generally.
			If $\underline{w}$ is a Coxeter word, we write $\Roudg{\underline{w}}$ for the corresponding positive Rouquier
			complex (we should replace $\underline{w}$ with its positive word lift). We now describe 
			a new complex $F_{\underline{w}}$ that we will obtain after Gaussian elimination.
			
			If $s\in S$ and $\underline{x}$ is the word consisting 
			of $n$ repetitions of the letter $s$, we set
			\[
				\worsum{\underline{x}}:=B_s(-n+1).
			\]
			If $\underline{x}$ is any Coxeter word, first write it as 
			\[
				\underline{x}_1\underline{x}_2\dots \underline{x}_k
			\]
			where each $\underline{x}_i$ is monotonous but $\underline{x}_i\underline{x}_{i+1}$ is not.
			Then set
			\[
				\worsum{\underline{x}}:=\worsum{\underline{x}_1}\worsum{\underline{x}_2}\cdots \worsum{\underline{x}_k}\underset{\oplus}{\subset} B_{\underline{x}}.
			\]
			We also set $\worsum{\emptyset}:=\mathbbm{1}$.
			\begin{exa}
				Let $\underline{x}=sssttusuu$, then 
				\begin{align*}
					&\underline{x}_1=sss,& &\underline{x}_2=tt,& &\underline{x}_3=u,& &\underline{x}_4=s,& &\underline{x}_5=uu.
				\end{align*}
				Hence $\worsum{\underline{x}}=B_s(-2)\otimes B_t(-1) \otimes B_u \otimes B_s \otimes B_u(-1)=B_{\underline{stusu}}(-4)$.
			\end{exa}
			Let now $\underline{w}$ be a Coxeter word:
			the graded object corresponding to $F_{\underline{w}}$ is
			\[
				\grad{(F_{\underline{w}})}=\bigoplus_{\underline{x}\preceq \underline{w}} \worsum{\underline{x}} \Tate{q_{\underline{w},\underline{x}}},
			\]
			where $q_{\underline{w},\underline{x}}=\ell(\underline{w})-\ell(\underline{x})$.
		
			We have to describe the differential map $d$.
			We will specify the components 
			\[
				d_{\underline{x}',\underline{x}}:\worsum{\underline{x}}\Tate{q_{\underline{w},\underline{x}}}\rightarrow \worsum{\underline{x}'}\Tate{q_{\underline{w}',\underline{x}'}},
			\]
			for $\underline{x},\underline{x}'\preceq \underline{w}$ and $\ell(\underline{x})=\ell(\underline{x}')+1$. 			
			This map is nonzero only if $\underline{x}'\preceq \underline{x}$, which means that $\underline{x}'$ is obtained 
			from $\underline{x}$ by eliminating one letter.
			The simplest such situation is when $\underline{x}$ and $\underline{x}'$ are monotonous of color $s$, of lengths $k+1$
			and $k$ respectively. Then we set
			\[
				d_{\underline{x}',\underline{x}}=d_{s,k}:=
				\begin{cases}
					\def.32cm{.32cm}
					\,\,
					\begin{tikzpicture}
						\pic[red] at (0,0) {dot=.32cm};
					\end{tikzpicture} & \text{if $k=0$,}\\
					\begin{tikzpicture}[baseline=.6*.32cm]
						\draw[red] (0,0)--(0,1.7*.32cm);
						\node[anchor=east] at (0,.85*.32cm) {$\delta_{\re{s}}$};
					\end{tikzpicture}
					\hphantom{\delta_s}
					-
					\hphantom{\delta_s}
					\begin{tikzpicture}[baseline=.6*.32cm]
						\draw[red] (0,0)--(0,1.7*.32cm);
						\node[anchor=west] at (0,.85*.32cm) {$\delta_{\re{s}}$};
					\end{tikzpicture} 
					=\,
					\begin{tikzpicture}[baseline=.6*.32cm]
						\draw[red] (0,0)--(0,1.7*.32cm);
						\node[anchor=east] at (0,.85*.32cm) {$\alpha_{\re{s}}$};
					\end{tikzpicture}
					\hphantom{\delta_s}- \hphantom{\delta_s}
					\begin{tikzpicture}[baseline=.6*.32cm]
						\draw[red] (0,0)--(0,.6*.32cm);\fill[red] (0,0.6*.32cm) circle (1.5pt);
						\draw[red] (0,1.7*.32cm)--(0,1.1*.32cm);\fill[red] (0,1.1*.32cm) circle (1.5pt);
					\end{tikzpicture} & \text{if $k>0$ is odd,}
					\\[1em]
					\begin{tikzpicture}[baseline=.6*.32cm]
						\draw[red] (0,0)--(0,1.7*.32cm);
						\node[anchor=east] at (0,.85*.32cm) {$\delta_{\re{s}}$};
					\end{tikzpicture}
					\hphantom{\delta_s}-\hphantom{\delta_s}
					\begin{tikzpicture}[baseline=.6*.32cm]
						\draw[red] (0,0)--(0,1.7*.32cm);
						\node[anchor=west] at (0,.85*.32cm) {${\re{s}}(\delta_{\re{s}})$};
					\end{tikzpicture}
					=\,
					\begin{tikzpicture}[baseline=.6*.32cm]
						\draw[red] (0,0)--(0,.6*.32cm);\fill[red] (0,0.6*.32cm) circle (1.5pt);
						\draw[red] (0,1.7*.32cm)--(0,1.1*.32cm);\fill[red] (0,1.1*.32cm) circle (1.5pt);
					\end{tikzpicture} & \text{if $k>0$ is even.}
				\end{cases}  			
			\]
			In general, one can write $\underline{x}$ and $\underline{x}'$ 
			in the form
			\begin{align*}
				&\underline{x}=\underline{z}_1 \underbrace{sss\dots s}_{k+1} \underline{z}_2,& 
				&\underline{x}'=\underline{z}_1 \underbrace{ss\dots s}_{k} \underline{z}_2,
			\end{align*}
			where $\underline{z}_1$ and $\underline{z}_2$ are (possibly empty) Coxeter words such that $\underline{z}_1$ does not
			end with $s$ and $\underline{z}_2$ does not start with $s$. Hence we set
			\[
				d_{\underline{x}',\underline{x}}:=(-1)^{\ell(\underline{z}_1)} \id_{\worsum{\underline{z}_1}}\otimes d_{s,k} \otimes \id_{\worsum{\underline{z}_2}}.
			\]
			If $k=0$ and $\underline{z}_1$ ends with same letter as $\underline{z}_2$ starts, then we also compose on top 
			with the corresponding trivalent:
			\[
				\begin{tikzpicture}[y=.3cm,x=.4cm,baseline=.5cm]
					\draw[gray] (-1,0)--(7,0);
					\draw[blue] (2,0)--(3,2)--(4,0);\draw[blue](3,2)--(3,4);
					\draw[violet] (1,0)--(1,4);\draw[violet] (5,0)--(5,4);
					\pic[red] at (3,0) {dot={1cm}};
					\node at (0,2) {\dots};\node at (6,2) {\dots};
				\end{tikzpicture}			
			\]
			\begin{exa}
				Let $\underline{w}=\bl{t}\re{ss}\bl{t}$, then $F_{\underline{w}}$ is
				\[
					\begin{tikzcd}[row sep=tiny]
																				& 															& \worsum{\underline{tt}}(2)\ar[rdd]			& 						&						\\
																				& \worsum{\underline{tst}}(1)\ar[ru,"d_2"]\ar[rd]\ar[rddd]	& \oplus										&						&						\\
																				& \oplus													& \worsum{\underline{ts}}(2)\ar[r]\ar[rdd]		& \worsum{t}(3)\ar[rd]	&						\\
						\worsum{\underline{tsst}}\ar[ruu,"d_1"]\ar[r]\ar[rdd]	& \worsum{\underline{tss}}(1)\ar[ru]\ar[rddd]				& \oplus 										& \oplus				& \worsum{\emptyset} 	\\
																				& \oplus													& \worsum{\underline{st}}(2)\ar[r]\ar[ruu]		& \worsum{s}(3)\ar[ru]	& 						\\
																				& \worsum{\underline{sst}}(1)\ar[ru]\ar[rd]					& \oplus										& 						&						\\
																				& 															& \worsum{\underline{ss}}(2)\ar[ruu]			& 						&
					\end{tikzcd}
				\]
				We describe the arrows $d_1$ and $d_2$, as an example, and we leave the others to the reader.
				We have $\worsum{\underline{tsst}}=B_tB_sB_t(-1)$ and $\worsum{\underline{tst}}(1)=B_tB_sB_t(1)$. We are canceling the letter $s$
				and passing from two occurrences to one. Furthermore the length of the word preceding the
				group of $s$'s is one, hence the morphism $d_1$ is:
				\[	
					-
					\begin{tikzpicture}[x=.5cm,baseline=.4cm]
						\draw[gray] (0.5,0)--(3.5,0); \draw[gray] (.5,1)--(3.5,1);
						\draw[blue] (1,0)--(1,1); \draw[blue] (3,0)--(3,1);
						\draw[red] (2,0)--(2,1);
						\node at (1.5,0.5) {$\delta_{\re{s}}$};
					\end{tikzpicture}
					+
					\begin{tikzpicture}[x=.5cm,baseline=.4cm]
						\draw[gray] (0.5,0)--(3.5,0); \draw[gray] (.5,1)--(3.5,1);
						\draw[blue] (1,0)--(1,1); \draw[blue] (3,0)--(3,1);
						\draw[red] (2,0)--(2,1);
						\node at (2.5,0.5) {$\delta_{\re{s}}$};
					\end{tikzpicture}
				\]
				Next, we have $\worsum{\underline{tt}}(2)=B_t(1)$ and the morphism $d_2$ is eliminating the only 
				$s$ from $\worsum{\underline{tst}}$. Notice that the two adjacent letters are both $t$'s, hence $d_2$ is
				\[	
					-
					\begin{tikzpicture}[x=.4cm,y=.5cm,baseline=.3cm]
						\draw[gray] (0.5,0)--(3.5,0); \draw[gray] (0.5,2)--(3.5,2);
						\draw[blue] (1,0)--(2,1)--(2,2); \draw[blue] (3,0)--(2,1);
						\draw[red] (2,0)--(2,0.5);\fill[red] (2,0.5) circle (1.5pt);
					\end{tikzpicture}
				\]
			\end{exa}
			\begin{pro}\label{pro_dsquared0}
				The differential map $d$ defined above satisfies $d^2=0$.
			\end{pro}
			One can prove this directly but it will follow from Claim \ref{cla_morphcomplex} below
			(which is proven independently of this statement).

			Here is the statement of the main result of this section.
			\begin{thm}\label{thm_reduce}
				Let $\underline{w}$ be a Coxeter word. Then the complex $\Roudg{\underline{w}}$ admits
				$F_{\underline{w}}$ as a Gaussian summand. In particular $\Roudg{\underline{w}} \simeq F_{\underline{w}}$.
			\end{thm}
			\begin{exa}
				If $\underline{w}=\underline{s}^n$, the theorem gives the minimal complex for $F_s^n$. See
				\cite[Exercice 19.29]{EMTW}.
			\end{exa}
			The rest of this section is devoted to the proof of this result.
			We will explicitely construct a projection from $\Roudg{\underline{w}}$ to $F_{\underline{w}}$ and then show 
			that it is a homotopy equivalence.
			\subsection{Multiword decomposition}\label{subs_extrem}
			The first step is a convenient decomposition of the original complex, generalizing that 
			of Example \ref{exa_sss}.
			We will need the following notion.
			
			We call \emph{multiword} an expression of the form
			\[
				\mu=s_1^{n_1} \dots s_k^{n_k}
			\]
			with $s_i\in S$ and $n_i\in \mathbb{Z}_{>0}$. Notice that $s_i$ need not be different from $s_{i+1}$. 
			If all the $s_i$ are equal we still say that $\mu$ is \emph{monotonous}.
			Equivalently, a multiword is an element of the free monoid 
			generated by $S\times \mathbb{Z}_{>0}$. Let $\mathfrak{M}$ denote the set of multiwords.
			To each multiword we can associate a Coxeter word in two natural ways: one can either
			expand the exponents, or forget them. This gives two maps 
			\[	
				\begin{tikzcd}
					\mathfrak{M} \ar[r,shift left,"e"]\ar[r,shift right,swap,"f"] & \mathfrak{W}_S,
				\end{tikzcd}
			\]
			where
			\begin{align*}
				& e(\mu)=\underbrace{s_1\dots s_1}_{n_1} \dots \underbrace{s_k\dots s_k}_{n_k}, \\
				& f(\mu)=s_1\dots s_k.			
			\end{align*}			
			We say that a multiword is \emph{simple} if all multiplicities are 1's, or equivalently
			$e(\mu)=f(\mu)$.
			
			Recall now the decomposition of Example \ref{exa_basicdeco}. 
			Iterating, we get
			\begin{align*}
				& B_s^{\otimes 3}= B_s(-2)\oplus B_s^{\oplus 2} \oplus B_s(2), \\
				& B_s^{\otimes 4}= B_s(-3)\oplus B_s(-1)^{\oplus 3}\oplus B_s(1)^{\oplus 3} \oplus B_s(3), \\
				& \dots,
			\end{align*}			
			and in general
			\[
				B_s^{\otimes (n+1)}=\bigoplus_{k=0}^{n} B_s(n-2k)^{\oplus {\binom{n}{k}}}.
			\]
			The inclusion morphisms of the $\binom{n}{k}$ summands of the form $B_s(n-2k)$
			are all possible combinations of inclusions $\iota_1$ and $\iota_2$ from
			Example \ref{exa_basicdeco}. Hence they are of the form
			\[
				\begin{tikzpicture}[x=.3cm,y=.7cm]
					\draw[red] (0,0)--(0,1);
					\draw[red] (-15,2) ..controls (-15,1).. (0,1) ..controls (15,1).. (15,2);
					\draw[decorate,decoration={brace}] (-15,2.2) to node[above] {$n_1$} (-9,2.2);
					\draw[decorate,decoration={brace}] (-7,2.2) to node[above] {$n_2$} (-1,2.2);
					\node at (4,2.3) {\dots};
					\draw[decorate,decoration={brace}] (9,2.2) to node[above] {$n_k$} (15,2.2);
					\clip (-15,2) ..controls (-15,1).. (0,1) ..controls (15,1).. (15,2);
					\foreach \i in {-13,-9,-7,-5,-1,1,3,7,9,11}{
						\draw[red] (\i,0)--(\i,2);
						}
					\foreach \i in {-8,0,8}{
						\node[anchor=south] at (\i,1) {$\delta_{\re{s}}$};
						}
					\foreach \i in {-11,-3,5,13}{
						\node[anchor=south,red] at (\i,1.5) {\dots};
						}
				\end{tikzpicture}
			\]
			where $n_1+n_2+\dots +n_k=n+1$, and there is no decoration in between each group of $n_i$ strands.
			We label these summands by monotonous multiwords: the summand corresponding to the inclusion above
			will be denoted by $C_{\mu}$ with $\mu=s^{n_1}s^{n_2}\dots s^{n_k}$.
			The corresponding projection is obtained by appropriately combining projections
			$\pi_1$ and $\pi_2$ from Example \ref{exa_basicdeco}: one reflects the diagram vertically,
			and replaces empty decorations with $-s(\delta_s)$ and $\delta_s$ with empty decorations.
			\begin{exa}
				There are $\binom{3}{2}=3$ copies of 
				$B_s(-1)$ inside $B_s^{\otimes 4}$. Their inclusion	mophisms are:
				\begin{align*}
					&\begin{tikzpicture}[scale=.35,baseline=1cm]
						\draw[red] (0,0)--(0,1)--(-3,4);
							\foreach \i in {0,1,2}{%
								\draw[red] ({-\i},{1+\i})--({3-2*\i},4);
								}
							\foreach \i in {1,2}{%
								\node[anchor=south, inner sep=.2cm] at ({-\i},{1+\i}) {$\delta_{\re{s}}$};
								}
					\end{tikzpicture}&
					&\begin{tikzpicture}[scale=.35,baseline=1cm]
						\draw[red] (0,0)--(0,1)--(-3,4);
							\foreach \i in {0,1,2}{%
								\draw[red] ({-\i},{1+\i})--({3-2*\i},4);
								}
							\foreach \i in {0,2}{%
								\node[anchor=south, inner sep=.2cm] at ({-\i},{1+\i}) {$\delta_{\re{s}}$};
								}
					\end{tikzpicture}&
					&\begin{tikzpicture}[scale=.35,baseline=1cm]
						\draw[red] (0,0)--(0,1)--(-3,4);
							\foreach \i in {0,1,2}{%
								\draw[red] ({-\i},{1+\i})--({3-2*\i},4);
								}
							\foreach \i in {0,1}{%
								\node[anchor=south, inner sep=.2cm] at ({-\i},{1+\i}) {$\delta_{\re{s}}$};
								}
					\end{tikzpicture}
				\end{align*}
				and the corresponding projection morphisms are respectively:
				\begin{align*}
					&\begin{tikzpicture}[scale=.35,yscale=-1,baseline=0]
						\draw[red] (0,0)--(0,1)--(-3,4);
							\foreach \i in {0,1,2}{%
								\draw[red] ({-\i},{1+\i})--({3-2*\i},4);
								}
							\foreach \i in {0}{%
								\node[anchor=north west, inner sep=.1cm,font=\small,rotate=-45] at ({-\i},{1+\i}) {$-\re{s}(\delta_{\re{s}})$};
								}
					\end{tikzpicture}&
					&\begin{tikzpicture}[scale=.35,yscale=-1,baseline=0]
						\draw[red] (0,0)--(0,1)--(-3,4);
							\foreach \i in {0,1,2}{%
								\draw[red] ({-\i},{1+\i})--({3-2*\i},4);
								}
							\foreach \i in {1}{%
								\node[anchor=north west, inner sep=.1cm,font=\small,rotate=-45] at ({-\i},{1+\i}) {$-\re{s}(\delta_{\re{s}})$};
								}
					\end{tikzpicture}&
					&\begin{tikzpicture}[scale=.35,yscale=-1,baseline=0]
						\draw[red] (0,0)--(0,1)--(-3,4);
							\foreach \i in {0,1,2}{%
								\draw[red] ({-\i},{1+\i})--({3-2*\i},4);
								}
							\foreach \i in {2}{%
								\node[anchor=north west, inner sep=.1cm,font=\small,rotate=-45] at ({-\i},{1+\i}) {$-\re{s}(\delta_{\re{s}})$};
								}
					\end{tikzpicture}
				\end{align*}
				With the multiword notation the three summands are, respectively,
				\begin{align*}
					&C_{sss^2}& &C_{ss^2s}& &C_{s^2ss}.				
				\end{align*}
			\end{exa}
			Let us now pass to the general case. For an arbitrary multiword
			$\mu$, first we decompose it into monotonous multiwords $\mu_1\dots\mu_k$
			with $\mu_i$ and $\mu_{i+1}$ of different colors, then we set
			\[
				C_{\mu}:=C_{\mu_1}\dots C_{\mu_k}.
			\]
			Notice that when the multiword $\mu$ is simple and $e(\mu)=f(\mu)=\underline{x}$ then
			$C_{\mu}=C_{\underline{x}}$, so our notation is consistent.
			
			Let $\underline{x}$ be an arbitrary Coxeter word.	We can apply the one color case to 
			each sequence of repeated letters and decompose the Bott-Samelson object $B_{\underline{x}}$ 
			accordingly. Hence we get 
			\begin{pro}\label{pro_multiwdec}
			Let $\underline{x}$ be a Coxeter word, then:
			\[
				B_{\underline{x}}=\bigoplus_{e(\mu)=\underline{x}} C_{\mu}.
			\]
			\end{pro}
			Let us apply this to the graded object $\grad{(\Roudg{\underline{w}})}$.
			For a subexpression $\mathbf{i}$ of $\underline{w}$, let $\underline{w}_{\mathbf{i}}$ be the subword of $\underline{w}$ corresponding 
			to it. For a given subword $\underline{x}$, let $M_{\underline{x}}$ denote the following set:
			\[
				M_{\underline{x}}:=\{ (\mathbf{i},\mu) \in  \{0,1\}^{\ell(\underline{w})}\times\mathfrak{M} \mid f(\mu)=\underline{x}, e(\mu)=\underline{w}_{\mathbf{i}}\}.
			\]
			Then, using Proposition \ref{pro_multiwdec}, we have:
			\[
				\grad{(\Roudg{\underline{w}})}=\bigoplus_{\mathbf{i}\in \{0,1\}^{\ell(\underline{w})}} B_{\mathbf{i}}\Tate{q_{\mathbf{i}}}=
				\bigoplus_{\mathbf{i}}\,\bigoplus_{e(\mu)=\underline{w}_{\mathbf{i}}} \mulsum{\mu}\Tate{q_{\mathbf{i}}}=
				\bigoplus_{\underline{x}\preceq \underline{w}}\,\,\underbrace{%
					\bigoplus_{(\mathbf{i},\mu)\in M_{\underline{x}}} \mulsum{\mu}\Tate{q_{\mathbf{i}}} }_{:=G_{\underline{x}}}.
			\]
			We will use the notation $C_{\mathbf{i},\mu}:=C_{\mu}\Tate{q_{\mathbf{i}}}$. 
			
			In Example \ref{exa_sss}, the subwords of $\underline{w}=sss$ are 
			\[
				\underline{sss},\underline{ss}, \underline{s}, \emptyset.
			\]
			So the corresponding sets are the following:
			\begin{align*}
				& M_{\underline{sss}} = \{ (111,sss) \}, \\
				& M_{\underline{ss}} = \{ (110,ss),(101,ss),(011,ss),(111,s^2s),(111,ss^2)\}, \\
				& M_{\underline{s}} = \{ (100,s),(010,s),(001,s),(110,s^2),(101,s^2),(011,s^2),(111,s^3)\}, \\
				& M_{\emptyset} = \{ (000,\emptyset) \}.
			\end{align*}
			Then $G_{\underline{sss}},G_{\underline{ss}},G_{\underline{s}}$ and $G_{\emptyset}$ are 
			respectively the violet, blue, green and black parts in \eqref{eq_exadecomcol}.			
			
			These $G_{\underline{x}}$ allow us to define a filtration of the complex as in the example.
			\begin{lem}\label{lem_filtr}
				Choose an total ordering $\underline{x}_0,\underline{x}_1,\dots, \underline{x}_N$ of the subwords of $\underline{w}$, refining
				$\preceq$.
				Then the objects 
				\[
					G_{\le n} := \bigoplus_{i=0}^n F_{\underline{x}_i}
				\]
				define a filtration of $\Roudg{\underline{w}}$ by subcomplexes.
			\end{lem}
			\begin{proof}
				It is sufficient to prove that the component $G_{\underline{x}}\rightarrow G_{\underline{z}}$ of the 
				differential map can be nonzero only if $\underline{z}=\underline{x}$ or $\underline{z} \preceq \underline{x}$. Consider
				the composition:
				\begin{equation}\label{eq_compdiff}
					\begin{tikzcd}
						\mulsum{\mathbf{i},\mu}\ar[r,hook]	& B_{\mathbf{i}}\ar[r,"d"]	& B_{\mathbf{i}'}\ar[r,two heads]		& \mulsum{\mathbf{i}',\mu'}.
					\end{tikzcd}				
				\end{equation}
				This is nonzero only if $\mathbf{i}'$ is obtained from $\mathbf{i}$ by turning a 1 to a 0, by definition
				of the differential. Then $e(\mu')\preceq e(\mu)$ and $\ell(e(\mu))=\ell(e(\mu'))+1$. Then 
				one can easily see that either $f(\mu')=f(\mu)$ or $f(\mu')\preceq f(\mu)$.
			\end{proof}
			As in Example \ref{exa_sss}, each of the subquotients $G_{\underline{x}}$ will shrink to a single object $C_{\underline{x}}$, via 
			large scale Gaussian elimination, and we will assemble these reductions together to a 
			reduction of the entire complex.
			\subsection{Colored polytopal complexes}
			To reduce the subquotients we will use large scale Gaussian elimination.
			More precisely, the subquotient $G_{\underline{x}}$ is left-described by a collapsible polytopal set
			$\Pi(\underline{w},\underline{x})$ that we now describe.
							
			Let $\Delta^i$ denote the $i$-simplex. The polytopes of $\Pi(\underline{w},\underline{x})$ 
			are labeled by the set $M_{\underline{x}}$.
			The polytope labeled $(\mathbf{i},\mu)$ is a multisimplex 
			$\Delta^{n_1-1}\times \dots \times \Delta^{n_k-1}$, where $\mu=s_1^{n_1}\dots s_k^{n_k}$.
			The faces of this multisimplex are those corresponding 
			to pairs $(\mathbf{i}',\mu')$, where $\mathbf{i}'\le \mathbf{i}$ (componentwise) and $\mu'$ is of the form $s_1^{m_1}\dots s_k^{m_k}$
			with $1\le m_i\le n_i$.			
			For instance, the vertices are labeled by all the subexpressions for $\underline{x}$ (and the only
			simple multiwords for them). 
			\begin{exa}
			Here are two examples of these polytopal sets. 
			\begin{center}
				\begin{tikzpicture}[xscale=-1,scale=.8,font=\footnotesize,baseline=-4cm]
					\coordinate (st0000) at (1,0); \coordinate (s00t00) at (0,-1);
					\coordinate (00st00) at (-1,-2); \coordinate (s0000t) at (1,-2);
					\coordinate (00s00t) at (0,-3); \coordinate (0000st) at (1,-4);
					\fill[green!50] (st0000)--(00st00)--(0000st)--cycle;				
					\draw (st0000)--(s00t00)--(s0000t)--(00s00t)--(0000st); 
					\draw (s00t00)--(00st00)--(00s00t);
					\draw(st0000)--(0000st);
					\foreach \i in {st0000,s00t00,s0000t,00st00,00s00t,0000st}{%
						\fill (\i) circle (1.2pt);
						}
					\node[anchor=south east] at (st0000) {\redblue{1}{1}{0}{0}{0}{0}};
					\node[anchor=south west] at (s00t00) {\redblue{1}{0}{0}{1}{0}{0}};
					\node[anchor=west] at (00st00) {\redblue{0}{0}{1}{1}{0}{0}};
					\node[anchor=east] at (s0000t) {\redblue{1}{0}{0}{0}{0}{1}};
					\node[anchor=north west] at (00s00t) {\redblue{0}{0}{1}{0}{0}{1}};
					\node[anchor=north east] at (0000st) {\redblue{0}{0}{0}{0}{1}{1}};
				\end{tikzpicture}
				\quad 
				\tdplotsetmaincoords{70}{93}
				\begin{tikzpicture}[tdplot_main_coords,scale=1.4,font=\footnotesize]\label{obelisque}
					\def 1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866{1} \def \s{.8pt} \def \f{\tiny} \def 100{0.866}
					\coordinate (A1) at (0,0,0); \coordinate(A2) at (0,1,0); \coordinate(A3) at (100,0.5,0); 
					\coordinate (B1) at (0,0,1); \coordinate(B2) at (0,1,1); \coordinate(B3) at (100,0.5,1);
					\coordinate (C1) at (0,0,2); \coordinate(C2) at (0,1,2); \coordinate(C3) at (100,0.5,2);
					\coordinate (D) at ({100/2},0.5,3);
					\fill[yellow!50] (D)--(C1)--(B1)--(A1)--(A3)--(B3)--(C3)--cycle;
					\fill[yellow!70] (D)--(C2)--(B2)--(A2)--(A3)--(B3)--(C3)--cycle;
					\draw(A3)--(A1); \draw(A2)--(A3); \draw[gray,dashed] (A1)--(A2);
					\draw(B3)--(B1); \draw(B2)--(B3); \draw[gray,dashed] (B1)--(B2);
					\draw(C3)--(C1); \draw(C2)--(C3); \draw[gray,dashed] (C1)--(C2);
					\draw(D)--(C3)--(B3)--(A3); 
					\draw(D)--(C2)--(B2)--(A2); 
					\draw(D)--(C1)--(B1)--(A1);
					\foreach \i in {A1,A2,A3,B1,B2,B3,C1,C2,C3,D}{%
						\fill (\i) circle (\s);
						}
					\node[anchor=south] at (D) {\ssstustu{0}{0}{0}{0}{0}{1}{1}{1}};
					\node[anchor=east] at (C1) {\ssstustu{0}{0}{1}{0}{0}{0}{1}{1}};
					\node[anchor=east] at (B1) {\ssstustu{0}{0}{1}{1}{0}{0}{0}{1}};
					\node[anchor=east] at (A1) {\ssstustu{0}{0}{1}{1}{1}{0}{0}{0}};
					\node[anchor=west] at (C2) {\ssstustu{0}{1}{0}{0}{0}{0}{1}{1}};
					\node[anchor=west] at (B2) {\ssstustu{0}{1}{0}{1}{0}{0}{0}{1}};
					\node[anchor=west] at (A2) {\ssstustu{0}{1}{0}{1}{1}{0}{0}{0}};
					\node[anchor=north] at (C3) {\ssstustu{1}{0}{0}{0}{0}{0}{1}{1}};
					\node[anchor=north] at (B3) {\ssstustu{1}{0}{0}{1}{0}{0}{0}{1}};
					\node[anchor=north] at (A3) {\ssstustu{1}{0}{0}{1}{1}{0}{0}{0}};
				\end{tikzpicture}
			\end{center}
			The first is $\Pi(\underline{ststst},\underline{st})$, 
			the second is $\Pi(\underline{ssstustu},\underline{stu})$.
			The vertices are labeled by colored subexpressions: we color the symbols of the 01-sequences 
			according to the corresponding elements in $S$.			
			\end{exa}
			\begin{pro}\label{pro_leftdescr}
				The polytopal set $\Pi(\underline{w},\underline{x})$ left-describes the subquotient $G_{\underline{x}}$.			
			\end{pro}
			\begin{proof}
				Of course, we associate $\mulsum{\mathbf{i},\mu}$ to the polytope labeled $(\mathbf{i},\mu)$. We need to check that 
				the nonzero morphisms correspond to the face relationship and that they are isomorphisms.
				
				Recall the proof of Lemma \ref{lem_filtr} and the composition \eqref{eq_compdiff}.
				When $f(\mu')=f(\mu)$, the only possibility is that 
				\begin{align*}
					&\mu=\dots s_{i-1}^{n_{i-1}} s_i^{n_i} s_{i+1}^{n_{i+1}}\dots,&
					&\mu'=\dots s_{i-1}^{n_{i-1}} s_i^{n_i-1} s_{i+1}^{n_{i+1}}\dots.
				\end{align*}
				and one can check that the composition is just the identity, up to sign.
			\end{proof}
			\begin{pro}\label{pro_collaps}
				The polytopal set $\Pi(\underline{w},\underline{x})$ is collapsible.
			\end{pro}			 
			\begin{proof}
				Consider the lexicographic order on 01-sequences and take the smallest. This corresponds to 
				a point and it is easy to see that there is only one maximal polytope containing it. Then we can collapse them. Then we can proceed by induction.
			\end{proof}
			To finally prove Theorem \ref{thm_reduce}, we have to glue all the reductions of the subquotients together


















































































































			\subsection{Proof of the theorem}\label{subs_proofofthm}
			We now define the homotopy equivalence from $F_{\underline{w}}^\bullet$ to $F_{\underline{w}}$.
			Given an element of $\{0,1\}^n$, we call a \emph{flip} the replacement of a subsequence
			``01'' with ``10'' or viceversa.
			For each subexpression $\mathbf{i}$ let $|\mathbf{i}|$ be the minimal number of flips that are necessary
			to turn $\mathbf{i}$ into the sequence $11\dots 100\dots 0$.
			Consider the maps
			\[
				\begin{cases}
					C_{\mathbf{i},\mu} \overset{(-1)^{|\mathbf{i}|}}{\longrightarrow} C_{\underline{x}} & \text{if $e(\mu)=f(\mu)=\underline{x}$} \\
					C_{\mathbf{i},\mu}\rightarrow 0 & \text{otherwise}
				\end{cases}			
			\]
			One can glue them together to form a morphism $\pi:\grad{(\Roudg{\underline{w}})}\rightarrow \grad{(F_{\underline{w}})}$.
			\begin{cla}\label{cla_morphcomplex}
				The morphism $\pi$ is compatible with the differential maps. 
			\end{cla}
			\begin{rmk*}
				 As $\pi$ is surjective, this also implies Proposition \ref{pro_dsquared0}.
			\end{rmk*}
			\begin{proof}[Proof of Claim \ref{cla_morphcomplex}]
				We will check this over each $\mulsum{\mathbf{i},\mu}$. Let first $\mu$ be simple and
				$e(\mu)=f(\mu)=\underline{x}$. Let $\underline{x}'$ be a word obtained by $\underline{x}$ by removing a letter. Then 
				the component $\mulsum{\mathbf{i},\mu}\rightarrow \worsum{\underline{x}'}$ of the composition $d\pi$ is 
				\[
					(-1)^{|\mathbf{i}|} d_{\underline{x}',\underline{x}}
				\]
				as defined in \S\,\ref{subs_reducedcom}. The same component of the composition $\pi d$
				is the sum of the compositions
				\begin{equation}\label{eq_compositionspid}
					\begin{tikzcd}
						\mulsum{\mathbf{i},\mu}\ar[r,hook]	& B_{\mathbf{i}}\ar[r,"d"]	& B_{\mathbf{i}'}\ar[r,two heads]		& \mulsum{\mathbf{i}',\mu'}\ar[r] & \worsum{\underline{x}'},
					\end{tikzcd}												
				\end{equation}
				for all possible $\mathbf{i}'$ obtained from $\mathbf{i}$ by turning a 1 to a 0. Let $\underline{x}=\underline{z}_1 sss\dots s\underline{z}_2$
				and $\underline{x}'=\underline{z}_1 ss\dots s\underline{z}_2$, then the only nonzero such composition are those where the new 
				$0$ corresponds to one of the $s$'s, and we get
				\[
					\sum_{\mathbf{i}'}(-1)^{|\mathbf{i}'|}
					\begin{tikzpicture}[x=.3cm,y=.4cm,baseline=.9cm]
						\draw[gray] (-12.5,0)--(12.5,0);\draw[gray] (-12.5,4)--(12.5,4);
						\draw[red] (0,0)--(0,1);
						\draw[red] (0,3)--(0,4);
						\draw[violet] (-10,0)--(-10,4); \draw[violet] (10,0)--(10,4);
						\node at (-11,2) {\dots};\node at (11,2) {\dots};
						\draw[red] (-9,2) ..controls (-9,1).. (0,1) ..controls (9,1).. (9,2)..controls (9,3).. (0,3) ..controls (-9,3).. (-9,2);
						\clip (-9,2) ..controls (-9,1).. (0,1) ..controls (9,1).. (9,2)..controls (9,3).. (0,3) ..controls (-9,3).. (-9,2);
						\foreach \i in {-7,-5,-1,3,7}{
							\draw[red] (\i,0)--(\i,4);
							}
						\draw[red](1,0)--(1,2); \fill[red] (1,2) circle (1.5pt);
						\foreach \i in {-8,-6,0,2,8}{
							\node[anchor=south] at (\i,1) {$\delta_{\re{s}}$};
							}
						\foreach \i in {-3,5}{
							\node[anchor=south,red] at (\i,1.5) {\dots};
							}						
					\end{tikzpicture}
				\]
				and one can check that this gives the same.
				
				If $\mu$ is not simple, we have to check that the composition $\pi d$ is zero.
				Consider again the component $\mulsum{\mathbf{i},\mu}\rightarrow \worsum{\underline{x}'}$, with 
				$\underline{x}'$ as above.
				The only non trivial case for the compositions \eqref{eq_compositionspid} is
				\begin{align*}
					&\mu=s_1 \dots s_{i-1} s_i^2 s_{i+1} \dots s_k,&
					&\mu=s_1 \dots s_{i-1} s_i s_{i+1} \dots s_k,&									
				\end{align*}
				and there are only two possibilities for $\mathbf{i}'$. These give
				\[
					\pm
					\begin{tikzpicture}[x=.6cm,y=.5cm,baseline=.4cm]
						\draw[gray] (-0.5,0)--(4.5,0);\draw[gray] (-0.5,2)--(4.5,2);
						\draw[violet] (1,0)--(1,2); \draw[violet] (3,0)--(3,2);
						\draw[red] (2,0)--(2,1); \draw[red] (2,1)--(1.75,1.5); \draw[red] (2,1)--(2.5,2);
						\node at (0,1) {\dots};\node at (4,1) {\dots};
						\fill[red] (1.75,1.5) circle (1.5pt);
					\end{tikzpicture}
					\mp
					\begin{tikzpicture}[x=.6cm,y=.5cm,baseline=.4cm]
						\draw[gray] (-0.5,0)--(4.5,0);\draw[gray] (-0.5,2)--(4.5,2);
						\draw[violet] (1,0)--(1,2); \draw[violet] (3,0)--(3,2);
						\draw[red] (2,0)--(2,1); \draw[red] (2,1)--(1.5,2); \draw[red] (2,1)--(2.25,1.5);
						\node at (0,1) {\dots};\node at (4,1) {\dots};
						\fill[red] (2.25,1.5) circle (1.5pt);
					\end{tikzpicture}
					=0
				\]
			\end{proof}				
			Now, consider a total ordering $\underline{x}_1,\dots \underline{x}_N$ on the subwords of $\underline{w}$.
			This gives a filtration on $\Roudg{\underline{w}}$ as in Lemma \ref{lem_filtr} and also naturally on 
			$F_{\underline{w}}$: the filtered pieces are just the $\worsum{\underline{x}_i}$.
			\begin{cla}
				The map $\pi$ is compatible with the filtrations. Furthermore, on subquotients, it coincides,
				up to a sign with the projection coming from Proposition \ref{pro_collaps}.
			\end{cla}
			\begin{proof}
				The compatibility with the filtrations is obvious by the definition of $\pi$.
				Notice that the edges of $\Pi(\underline{w},\underline{x})$ connect vertices that can be obtained 
				from one another by a \emph{colored flip}: this is a
				replacement of a subsequence
				``\textcolor{red}{1}0\dots 0\textcolor{red}{0}'' with ``\textcolor{red}{0}0\dots 0\textcolor{red}{1}'' 
				where the symbols in the middle are all zeros (that can have any color). 
				For example the following is a colored flip:
				\[
						\ssstustu{\underline{0}}{0}{0}{0}{0}{\underline{1}}{1}{1}\rightsquigarrow \ssstustu{\underline{1}}{0}{0}{0}{0}{\underline{0}}{1}{1}.
				\]
				Then one can check that the isomorphism associated to an edge is $(-1)^n$ where $n$ is 
				the number of zeros involved in the flip. In the example above $n=5$. Then one sees that 
				up to a global sign, the isomoprhisms associated with the paths to a given point agree with
				$\pi$.
			\end{proof}
			This concludes the proof of Theorem \ref{thm_reduce}.
			\subsection{Negative case}
			Of course everything works symmetrically for the negative case. Let $\underline{w}$ be a Coxeter word
			and $\underline{\omega}$ its negative word lift. Then one can describe a complex $F_{\underline{w}}^-$ homotopy equivalent to
			$\Roudg{\underline{\omega}}$ as follows.
			
			First we define, for $\underline{x}$ a word consisting of $n$ repetition of the letter $s$,
			\[
				\worsum{\underline{x}}^-:=B_s(n-1).
			\]
			Then, as before, with $\underline{x}=\underline{x}_1\underline{x}_2\dots \underline{x}_k$ as before, set
			\[
				\worsum{\underline{x}}:=\worsum{\underline{x}_1}^-\worsum{\underline{x}_2}^-\cdots \worsum{\underline{x}_k}^-\underset{\oplus}{\subset} B_{\underline{x}}.
			\]
			The graded object corresponding to $F_{\underline{w}}^-$ is 
			\[
				\grad{(F_{\underline{w}}^-)}=\bigoplus_{\underline{x}\preceq \underline{w}} \worsum{\underline{x}}^- \Tate{-q_{\underline{w},\underline{x}}}.
			\]
			The differential is given, again, via its components $d^-_{\underline{x}',\underline{x}}$. These
			are nonzero only if $\underline{x}'\succeq \underline{x}$ which implies that $\underline{x}'$ is obtained from $\underline{x}$
			by inserting a single occurrence of a letter $s$.
			Again, we first consider $\underline{x}$ and $\underline{x}'$ consisting of respectively $k$ and $k+1$ occurrences of the same letter $s$. 
			In this case
				\[
					d^-_{\underline{x}',\underline{x}}=d^-_{s,k}:=
					\begin{cases}
						\def.32cm{.32cm}
						\,\,
						\begin{tikzpicture}
							\pic[red,yscale=-1] at (0,0) {dot=.32cm};
						\end{tikzpicture} & \text{if $k=0$}\\
						\begin{tikzpicture}[baseline=.6*.32cm]
							\draw[red] (0,0)--(0,1.7*.32cm);
							\node[anchor=east] at (0,.85*.32cm) {$\delta_{\re{s}}$};
						\end{tikzpicture}
						\hphantom{\delta_s}-\hphantom{\delta_s}
						\begin{tikzpicture}[baseline=.6*.32cm]
							\draw[red] (0,0)--(0,1.7*.32cm);
							\node[anchor=west] at (0,.85*.32cm) {$\delta_{\re{s}}$};
						\end{tikzpicture}
						=\,
						\begin{tikzpicture}[baseline=.6*.32cm]
							\draw[red] (0,0)--(0,1.7*.32cm);
							\node[anchor=east] at (0,.85*.32cm) {$\alpha_{\re{s}}$};
						\end{tikzpicture}
						\hphantom{\delta_s}-\hphantom{\delta_s}
						\begin{tikzpicture}[baseline=.6*.32cm]
							\draw[red] (0,0)--(0,.6*.32cm);\fill[red] (0,0.6*.32cm) circle (1.5pt);
							\draw[red] (0,1.7*.32cm)--(0,1.1*.32cm);\fill[red] (0,1.1*.32cm) circle (1.5pt);
						\end{tikzpicture} & \text{if $k>0$ is odd}
						\\[1em]
						\begin{tikzpicture}[baseline=.6*.32cm]
							\draw[red] (0,0)--(0,1.7*.32cm);
							\node[anchor=east] at (0,.85*.32cm) {$\delta_{\re{s}}$};
						\end{tikzpicture}
						\hphantom{\delta_s}-\hphantom{\delta_s}
						\begin{tikzpicture}[baseline=.6*.32cm]
							\draw[red] (0,0)--(0,1.7*.32cm);
							\node[anchor=west] at (0,.85*.32cm) {${\re{s}}(\delta_{\re{s}})$};
						\end{tikzpicture}
						=\,
						\begin{tikzpicture}[baseline=.6*.32cm]
							\draw[red] (0,0)--(0,.6*.32cm);\fill[red] (0,0.6*.32cm) circle (1.5pt);
							\draw[red] (0,1.7*.32cm)--(0,1.1*.32cm);\fill[red] (0,1.1*.32cm) circle (1.5pt);
						\end{tikzpicture} & \text{if $k>0$ is even} 
					\end{cases}
				\]
				In general, one can write $\underline{x}$ and $\underline{x}'$ in the form
				\begin{align*}
					&\underline{x}=\underline{z}_1 \underbrace{\underline{ss\dots s}}_{k} \underline{z}_2,& 
					&\underline{x}'=\underline{z}_1 \underbrace{\underline{ss\dots s}}_{k+1} \underline{z}_2,&
				\end{align*}
				where $\underline{z}_1$ and $\underline{z}_2$ are as in the previous case. Then we set
				\[
					d_{\underline{x}',\underline{x}}:=(-1)^{|\underline{z}_1|} \id_{\mulsum{\underline{z}_1}^-}\otimes d^-_{s,k} \otimes \id_{\mulsum{\underline{z}_2}}.
				\]
				Again we need to treat the case when $k=0$ and $\underline{z}_1$ ends with same letter as 
				$\underline{z}_2$ starts. In this case we compose on bottom with a trivalent:
				\[
					\begin{tikzpicture}[y=.3cm,x=.4cm,baseline=.5cm]
						\draw[gray] (-1,0)--(7,0);
						\draw[blue] (2,4)--(3,2)--(4,4);\draw[blue](3,2)--(3,0);
						\draw[violet] (1,0)--(1,4);\draw[violet] (5,0)--(5,4);
						\pic[red,yscale=-1] at (3,4) {dot={1cm}};
						\node at (0,2) {\dots};\node at (6,2) {\dots};
					\end{tikzpicture}
				\]
				This concludes the definition of $F_{\underline{w}}^-$ and we have the analogue of Theorem
				\ref{thm_reduce}:
				\begin{thm*}
					Let $\underline{w}$ be a Coxeter word and $\underline{\omega}$ its negative word lift. Then the complex 
					$\Roudg{\underline{\omega}}$ admits $F_{\underline{w}}^-$ as a Gaussian summand. In particular 
					$\Roudg{\underline{\omega}} \simeq F_{\underline{w}}^-$.
				\end{thm*}
			For a arbitrary braid word $\underline{\omega}$ we can first write its alternating decomposition
			\[
				\underline{\omega}=\underline{\omega}_1\underline{\omega}_2\dots \underline{\omega}_k
			\]
			and consider the corresponding $F_{\underline{w}_i}$ and $F_{\underline{w}_j}^-$. Then define $F_{\underline{\omega}}$ to be their
			tensor product. Then we have
			$\Roudg{\underline{\omega}}\simeq F_{\underline{\omega}}$.
	\printbibliography[heading = bibintoc]
\end{document}