\section{Introduction}


Let $\Po$ be a finite partially ordered set (\textit{poset} in the sequel). All vector spaces in the current text are over a fixed algebraically closed field. Fixing an admissible dimension vector $\alv=(\alpha_0,\alpha_s)_{s\in \Po} \in \mathbb Z^{|\Po|+1}$ (see Preliminaries, for exact definition) consider a variety $\R{\alv}{\Po}$ of tuples of subspaces $(V_s)_{s\in \Po}$ of a vector space 
$V_0$ of dimension $\alpha_0$ such that $V_s\subseteq V_t,$ whenever $s\prec t$ in $\Po$ and $\dim V_s=\alpha_s$ for all $s$ in  $\Po$.
This variety was considered in \cite{futyus} as a set of all subspace representations of $\Po$ having dimension vector $\alv$. The interested reader is referred to \cite{NazarovaRoiter,simson} for information about representations of posets. The variety $\R{\alv}{\Po}$ 
can be viewed as a generalization of several important varieties. For instance: if $\Po$ is an anti-chain, then $\R{\alv}{\Po}$ is a product of Grassmannians; if $\Po$ is linearly ordered, then $\R{\alv}{\Po}$ is a partial flag variety; if $\Po$ is primitive (that is, a disjoint union of linearly ordered sets), then $\R{\alv}{\Po}$ is the multiple flag variety considered in \cite{mwz}. 

In this paper we give the dimension of $\R{\alv}{\Po}$ in terms of the Euler quadratic form $Q_\Po$ associated with $\Po$ (in particular, this answers the question posed in \cite{futyus}). Namely, we prove the following Tits-type equality:
\begin{thm*} 
For any finite poset $\Po$ and admissible vector $\alv=(\alpha_0,\alpha_s)_{s\in \Po} \in \mathbb Z^{|\Po|+1}$ one has
\begin{equation} \label{eqMain}
	\dim \mathrm{GL}(\alpha_0)-\dim \R{\alv}{\Po}=Q_\Po(\alv).
\end{equation}
\end{thm*}


The importance of integral bilinear forms and integral quadratic forms for classification problems of representations of posets and quivers was first observed in the works of Gabriel \cite{gabriel} and Drozd \cite{drozd} (see also  \cite{bongartz,simsonForms2,simsonForms} and references therein). For instance, the Tits quadratic form associated with a finite quiver has both geometric and homological interpretations (a generalization of the Tits quadratic form for quivers with relations was constructed in \cite{bongartz}). An important  role for applications in representation theory of posets is played by the Tits quadratic form and the Euler quadratic form (see, for instance, \cite{simsonForms2,simsonForms,simson}). The Tits quadratic form associated with a poset is known to have both geometric and homological interpretations 
(see, \cite{drozd}, \cite[Proposition 11.93]{simson} and \cite{bfksy} for an analogue of Tits quadratic form for unitary subspace representations of posets). 
It is well-known that Euler form associated with a poset has a homological interpretation, see, for instance, \cite[Lema 4.1]{simsonForms} and \cite{simsonForms2}. Our result can be viewed as a geometric interpretation of the Euler quadratic form. 

The paper is organized as follows. In Section 2 we give definitions and some preliminary results. In Section 3 we prove the main Theorem. In Section 4 we illustrate the proof of the main result by examples, and prove several consequences of the main result.


\section{Preliminaries}

A finite poset $\Po\equiv(\Po,\preceq)$ is given by the set of elements $\{s_1,\ldots,s_n\}$ and a partial order $\preceq$. Fixing the numbering in $\Po$ 
we make the identifications $\Mp{\Po}(\Zn)\equiv\Ma(\Zn)$ and $\Zn^{\Po}\equiv\Zn^{n}$. The relation $\preceq$ is uniquely defined by the \textit{incidence matrix} $\Mi{\Po}$ of $\Po$; that is, the integral square $n\times n$ matrix 
\begin{equation} \label{incidenceMatrix}
	\Mi{\Po} = [c_{st}]_{s,t \in \Po}\in \Mp{\Po}(\Zn), \quad \mbox{with}\ c_{st}=\left \{ \begin{array}{c}
	   1,\quad \mbox{for}\ s\preceq t,\\
	   0,\quad \mbox{for}\ s\npreceq t.\\ 
	\end{array} \right.
\end{equation}
Given a poset $\Po$, by $\Poe$ we understand its enlargement by unique maximal element $0$; that is, $\Poe\equiv(\Poe,\preceq^0)$ with $\Poe \setminus \{0\}=\Po$ and the order $\preceq^0$ is  obvious. 
 
Recall that the \textit{height} $h(\Po)$ is defined as the cardinality of the longest chain in $\Po$. By $\DD{s}$ denote the following set
$\DD{s}=\{t\in \Po \ | \ t\prec s\}$. The following proposition is a variation of the well-known Theorem (e.g., \cite[Theorem 2]{mir} for the original formulation and the proof).

\begin{prop}\label{partition}
Let $\Po$ be any poset. There are uniquely defined 
subsets $\T_i \subset \Po, i\in [1,h(\Po)]$ such that $\Po = \T_{h(\Po)} \sqcup \dots \sqcup \T_1$ and:
\begin{enumerate}[(i)]
\item for any $i,j$ with $i<j$ and any $t\in \T_i$, there is $s\in \T_j$ such that $t\prec s$ in $\Po$;
\item if $(r,s) \in \T_i \times \T_j$ and $r\prec s$ in $\Po$, then $i<j$. 
\end{enumerate}
\end{prop}

Let $\mathcal X,\mathcal Y$ be any subsets of $\Po$. By
$E_{\mathcal X}$ denote the column matrix $\begin{bmatrix}1 & 1 & \dots & 1 \end{bmatrix}^{tr}$ of size $|\mathcal{X}| \times 1$ and by
$\Mi{\mathcal X,\mathcal Y}$ denote the corresponding restriction of $\Mi{\Po}$; that is, $\Mi{\mathcal X,\mathcal Y}=[c_{st}]$ is an integral $|\mathcal X|\times |\mathcal Y|$ matrix, where $c_{st}$ is defined by \eqref{incidenceMatrix} with $s\in \mathcal X$ and $t\in \mathcal Y$. 
To simplify the notation we denote $\Mi{\mathcal X,\mathcal X}$ by $\Mi{\mathcal X}$, $\Mi{\T_i,\T_j}$ by $\Mi{i,j}$ and $E_{\T_i}$ by $E_i$. In particular, it is easy to see that $\Mi{i}$ is the identity matrix of size $|\T_i|$ (as each $\T_i$ is an antichain in $\Po$) which we denote by $\ID_i$. For any $k\in [1,h(\Po)]$ denote by $\Po_k$ the poset $\T_{h(\Po)} \sqcup \dots \sqcup \T_k$. In particular, $\Po_{1}=\Po$. If $\alv\in \Zn^{\Po}$, by ${\alv}_{\mathcal X}$ we denote the restriction of $\alv$ to $\mathcal X \subseteq \Po$. Again, to simplify the notation, ${\alv}_i$ denotes ${\alv}_{\T_i}$.

We will treat $\Mi{\Po}$ and $\alv\in \Zn^{\Po}$ with respect to the partition of $\Po$ as in Proposition \ref{partition}. That is, 
\begin{equation*}
\Mi{\Po}=
\begin{bmatrix}
\ID_{h(\Po)}& 0 & \dots & 0 \\
\Mi{h(\Po)-1,h(\Po)} & \ID_{h(\Po)-1} & \dots & 0 \\
\vdots &  \vdots & \ddots & \vdots \\
\Mi{1,h(\Po)} & \Mi{1,h(\Po)-1} & \dots &  \ID_{1}
\end{bmatrix},\quad 
\alv=(\alv_{h(\Po)},\dots,\alv_{1}).
\end{equation*}
It is easily verified and well-known that $\Mi{\Po}$ has the following decomposition
\begin{equation*}
\Mi{\Po}=F_{h(\Po)-1}\cdot \ldots \cdot F_{1},
\end{equation*}
in which each $F_{i}, i\in [1,h(\Po)-1]$ is a Frobenious-like matrix having the form
\begin{equation*}
F_i=
\begin{bmatrix}
\ID_{h(\Po)}& 0 & \dots & 0 & 0 & 0 &  \dots & 0 \\
0 & \ID_{h(\Po)-1} & \dots & 0 & 0 & 0 & \dots & 0\\
\vdots &  \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & \dots & \ID_{i+1} & 0 & 0 & \dots & 0 \\ 
\Mi{i,h(\Po)} & \Mi{i,h(\Po)-1} & \dots & \Mi{i,i+1}& \ID_{i} & 0 & \dots & 0 \\
0 & 0 & \dots & 0 & 0 & \ID_{i-1 }& \dots & 0 \\
\vdots &  \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & \dots &  0 & 0 & 0 & \dots & \ID_{1}
\end{bmatrix}.
\end{equation*}
Therefore 
$\Mi{\Po}^{-1}=F_{1}^{-1}\cdot \ldots \cdot F_{h(\Po)-1}^{-1}$, with
\begin{equation*}
F_i^{-1}=
\begin{bmatrix}
\ID_{h(\Po)}& 0 & \dots & 0 & 0 & 0 &  \dots & 0 \\
0 & \ID_{h(\Po)-1} & \dots & 0 & 0 & 0 & \dots & 0\\
\vdots &  \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & \dots & \ID_{i+1} & 0 & 0 & \dots & 0 \\ 
-\Mi{i,h(\Po)} & -\Mi{i,h(\Po)-1} & \dots & -\Mi{i,i+1}& \ID_{i} & 0 & \dots & 0 \\
0 & 0 & \dots & 0 & 0 & \ID_{i-1 }& \dots & 0 \\
\vdots &  \vdots & \ddots & \vdots & \vdots & \vdots & \ddots & \vdots\\
0 & 0 & \dots &  0 & 0 & 0 & \dots & \ID_{1}
\end{bmatrix}.
\end{equation*}


We say that $\alv=(\alpha_0,\alpha_s)_{s\in \Po}\in \Zn^{\Poe}$ is an \textit{admissible dimension vector} if $\alv_{\Po}\cdot \Mi{\Po}^{-1}$ is non-negative vector and $\alpha_0\geq \alpha_s$ for all $s\in \Po$. By $\A{\Po}$ denote the convex cone of all admissible dimension vectors. The following definition will be useful in the sequel. For any $\alv\in \Zn^{\Po}$ and $k\in [1,h(\Po)]$ define 
\begin{equation} \label{iterations}
\alv^{(k)}:=\alv^{(k-1)}\cdot F_{k-1}^{-1}, \quad \mbox{with}\quad \alv^{(1)}:=\alv. 
\end{equation}
Hence, it follows that 
\begin{equation} \label{iterCartan}
\alv\cdot \Mi{\Po}^{-1}=\alv\cdot F_{1}^{-1}\cdot \ldots \cdot F_{h(\Po)-1}^{-1}=\alv^{(h(\Po))},
\end{equation}
and  that $\alv_{\Po}\in \A{\Po}$ iff all vectors $\alv^{(k)}$ are non-negative.


We denote by $\QT{\Po},\Q{\Po}:\Zn^{\Poe} \to \Zn$  the Tits quadratic form, and the Euler quadratic form of ${\Poe}$ defined by the formula
\begin{equation*}
\begin{split}
\QT{\Po}(\alv)&=\alv \cdot \Ti{\Po} \cdot \alv^{tr}, \\
\Q{\Po}(\alv)&=\alv \cdot \Mi{\Poe}^{-1} \cdot \alv^{tr},
\end{split}
\end{equation*}
respectively, where 
$\Ti{\Po}=\left[
\begin{array}{c|c}
  1 & 0 \\ \hline
  -E_\Po & \Mi{\Po}^{tr}
\end{array}
\right]\in \Mp{\Poe}(\Zn)$
is the Tits matrix of $\Poe$. The reader is referred to \cite{simsonForms,simson} (and references therein) for a detailed study of quadratic and bilinear forms associated with posets and their applications for classification problems of representations of posets. 


\section{Proof of the Theorem}

First we prove two auxiliary lemmas. Denote by $d_{\Po}$ the function $\R{\alv}{\Po}\to \mathbb Z$ which maps each point $(V_s)_{s\in \Po}\in \R{\alv}{\Po}$ to $\dim\Big (\sum_{y\in \T_{h(\Po)}} V_y\Big )$. It is easy to check that $d_{\Po}$ is semi-continuous. Moreover we have the following 

\begin{lem} \label{sum}
Let $\alv\in \Zn^{\Poe}$ be a vector such that $\alv\cdot \Mi{\Poe}^{-1}$ is non-negative.
There is an open subset in $\R{\alv}{\Po}$ where $d_{\Po}$ attains maximum equal to 
$\alv_{\Po}\cdot \Mi{\Po}^{-1}\cdot E_\Po$. 
\end{lem}

\begin{proof}
First suppose that $h(\Po)=1$. In this case $\R{\alv}{\Po}=\prod_{s\in \Po} \Gr(\alpha_s,\alpha_0)$. As  $\alv\cdot \Mi{\Poe}^{-1}$ is non-negative for $\alv=(\alpha_0;\alpha_s)_{s\in \Po}$, we have that $\sum_{s\in \Po} \alpha_s\leq \alpha_0$. Hence for general point $(V_s)_{s\in \Po}$ in $\R{\alv}{\Po}$ the sum $\sum_{s\in \Po} V_s$ is direct, therefore $\dim(\sum_{s\in \Po} V_s)=\sum_{s\in \Po}\alpha_{s}$ is a maximal value of $d_\Po$, which is equivalent to statement. 

Assume that $h(\Po)>1$. Consider the natural map $\pi:\R{\alv}{\Po} \to \prod_{s\in \T_1} \Gr(\alpha_s,\alpha_0)$ which maps the tuple $(V_s)_{s\in \Po}$ to $(V_s)_{s\in \T_1}$. The map $\pi$ is obviously surjective (as $\T_1$ is a set of minimal points in $\Po$).
Recall that $\Po_2=\T_{h(\Po)}\sqcup\dots \sqcup \T_{2}$. Abusing the notation let $\alv^{(2)}$ be a restriction of $\alv^{(2)}$ (defined by \eqref{iterations}) to $\Po_2$. Consider the variety  $\R{\alv^{(2)}}{\Po_2}$ and the corresponding function $d_{\Po_2}$. Analyzing the fibers of the map $\pi$ it is straightforward to check that 
$$
	\max d_{\Po}=\max d_{\Po_2}+\alv_1\cdot E_1.
$$
Therefore, proceeding inductively, we get
$$
\max d_{\Po}=\sum_{i=1}^{h(S)} \alv_i^{(i)} \cdot E_i=\alv^{(h(S))}_{\Po}\cdot E_{\Po}.
$$
By \eqref{iterCartan} we have $\alv^{(h(S))}\cdot E_{\Po}=\alv_{\Po}\cdot \Mi{\Po}^{-1}\cdot E_\Po$, thus the statement follows. \end{proof}

\begin{lem} \label{lem_alg}
Let $\Po$ be a poset, $x$ be a maximal element in $\Po$ and $\alv \in \Zn^{\Poe}$. Then $$\Q{\Po}(\alv) - \Q{\Po\setminus\{x\}}(\alv_{\Po\setminus\{x\}}) = - (\alpha_x-{\alv}_{\DD{x}}\cdot\Mi{\DD{x}}^{\,-1}\cdot{E_{\DD{x}}})(\alpha_0-\alpha_x).$$ 
\end{lem}
\begin{proof}
If $h(\Po)=1$, then the statement is clear (since $\DD{x}$ is empty and $\Mi{\Po}^{\,-1} =\mathrm{Id}$).
Suppose $h(\Po)\geq 2$. Write $\Po$ as $\Jx \cup \{x\} \cup \DD{x}$ and 
$\alv$ as $(\alv_0;\alv_{\Jx},\alpha_x, \alv_{\DD{x}})$, where $\Jx=\Po \setminus (\DD{x} \cup \{x\})$.   The incidence matrices of $\Poe$ and $\Poe\setminus\{x\}$ (with respect to above decomposition of $\Po$) have the form
\begin{equation*}
  \Mi{\Poe} = \begin{bmatrix} 
    			1 & 0 & 0 & 0\\
                E_{\Jx} & \Mi{\Jx} & 0 & 0\\
                1 & 0 & 1 & 0\\
                E_{\DD{x}} & \Mi{\DD{x},\Jx} & E_{\DD{x}} & \Mi{\DD{x}}
    		   \end{bmatrix}, \quad 
\Mi{\Poe\setminus\{x\}} = \begin{bmatrix} 
    			1 & 0 & 0 \\
                E_{\Jx} & \Mi{\Jx} & 0\\
                E_{\DD{x}} & \Mi{\DD{x},\Jx} & \Mi{\DD{x}}
    		   \end{bmatrix}.     
\end{equation*}
Therefore 
\begin{equation*}
\begin{split}
  \Mi{\Poe}^{\,-1}&=\begin{bmatrix} 
    			1 & 0 & 0 & 0\\
                -\Mi{\Jx}^{\,-1}E_{\Jx} & 
                \Mi{\Jx}^{\,-1} & 
                0 & 0\\
                -1 & 0 & 1 & 0\\
                \Mi{\DD{x}}^{\,-1}\Mi{\DD{x},\Jx}\Mi{\Jx}^{\,-1}E_{\Jx} & 
                 -\Mi{\DD{x}}^{\,-1}\Mi{\DD{x},\Jx}\Mi{\Jx}^{\,-1}& 
                -\Mi{\DD{x}}^{\,-1}E_{\DD{x}}& 
                \Mi{\DD{x}}^{\,-1}
    		   \end{bmatrix},\\
\Mi{\Poe\setminus\{x\}}^{\,-1}&=\begin{bmatrix} 
    			1 & 0 & 0 \\
                -\Mi{\Jx}^{\,-1}E_{\Jx} & 
                \Mi{\Jx}^{\,-1} & 0 \\
                \Mi{\DD{x}}^{\,-1}\Mi{\DD{x},\Jx}\Mi{\Jx}^{\,-1}E_{\Jx} - \Mi{\DD{x}}^{\,-1}E_{\DD{x}} & 
                -\Mi{\DD{x}}^{\,-1}\Mi{\DD{x},\Jx}\Mi{\Jx}^{\,-1}&         
                \Mi{\DD{x}}^{\,-1}
    		   \end{bmatrix}.               
\end{split}
\end{equation*}
Hence the corresponding Euler forms are:
\begin{equation*}
\begin{split}
\Q{\Po}(\alv) &=(\alv_0;\alv_{\Jx},\alpha_x, \alv_{\DD{x}})\cdot \Mi{\Poe}^{\,-1} \cdot (\alv_0;\alv_{\Jx},\alpha_x, \alv_{\DD{x}})^{tr}\\
&= (\alpha_0 - \alv_{\Jx}\cdot\Mi{\Jx}^{\,-1}\cdot E_{\Jx}-\alpha_x + \alv_{\DD{x}}\cdot \Mi{\DD{x}}^{\,-1}\cdot\Mi{\DD{x},\Jx}\cdot\Mi{\Jx}^{\,-1})\alpha_0
\\&\qquad+(\alv_{\Jx}\cdot\Mi{\Jx}^{\,-1} - \alv_{\DD{x}}\cdot\Mi{\DD{x}}^{\,-1}\cdot\Mi{\DD{x},\Jx}\cdot\Mi{\Jx}^{\,-1})\cdot\alv_{\Jx}^{tr}
\\&\qquad+(\alpha_x - \alv_{\DD{x}}\cdot\Mi{\DD{x}}^{\,-1}\cdot E_{\DD{x}})\alpha_x+\alv_{\DD{x}}\cdot\Mi{\DD{x}}^{\,-1}\cdot\alv_{\DD{x}}^{tr},
\end{split}
\end{equation*}
and
\begin{equation*}
\begin{split}
&\Q{\Po\setminus\{x\}}(\alv_{\Po\setminus\{x\}})= (\alv_0;\alv_{\Jx},\alv_{\DD{x}})\cdot \Mi{\Poe\setminus\{x\}}^{\,-1} \cdot (\alv_0;\alv_{\Jx}, \alv_{\DD{x}})^{tr}\\ 
&\quad= (\alpha_0 - \alv_{\Jx}\cdot\Mi{\Jx}^{\,-1}\cdot E_{\Jx}+ \alv_{\DD{x}}\cdot\Mi{\DD{x}}^{\,-1}\cdot\Mi{\DD{x},\Jx}\cdot\Mi{\Jx}^{\,-1} -\alv_{\DD{x}}\cdot\Mi{\DD{x}}^{-1}\cdot E_{\DD{x}})\alpha_0
\\&\qquad +(\alv_{\Jx}\cdot\Mi{\Jx}^{\,-1} - \alv_{\DD{x}}\cdot\Mi{\DD{x}}^{\,-1}\cdot\Mi{\DD{x},\Jx}\cdot\Mi{\Jx}^{\,-1})\cdot\alv_{\Jx}^{tr}+\alv_{\DD{x}}\cdot\Mi{\DD{x}}^{\,-1}\cdot\alv_{\DD{x}}^{tr}.
\end{split}
\end{equation*}
Thus, 
\begin{equation*}
\begin{split}
\Q{\Po}(\alv) - \Q{\Po\setminus\{x\}}(\alv_{\Po\setminus\{x\}}) &= -\alpha_x\alpha_0 + \alv_{\DD{x}}\cdot\Mi{\DD{x}}^{-1}\cdot E_{\DD{x}}\alpha_0 + \alpha_x^2-\alv_{\DD{x}}\cdot\Mi{\DD{x}}^{\,-1}\cdot E_{\DD{x}}\alpha_x 
\\&= - (\alpha_x-{\alv}_{\DD{x}}\cdot\Mi{\DD{x}}^{\,-1}\cdot{E_{\DD{x}}})(\alpha_0-\alpha_x).
\end{split}
\end{equation*}
\end{proof}

Now we prove the main Theorem. We proceed by induction on cardinality of poset.
If $\Po=\{x\}$, then $\dim \R{\alv}{\Po} = \dim \Gr(\alpha_x,\alpha_0) = \alpha_x(\alpha_0-\alpha_x) $. On the other hand, 
    \begin{eqnarray*}
    	\dim \mathrm{GL}(\alpha_0)-\Q{\Po}(\alv) &=& {\alpha_0}^2 -\begin{bmatrix}\alpha_0 &\alpha_x \end{bmatrix}\cdot \begin{bmatrix}1 & 0\\ 1 & 1\end{bmatrix}^{\,-1} \cdot\begin{bmatrix}\alpha_0 \\\alpha_x \end{bmatrix} 
    \\&=& {\alpha_0}^2 - (\alpha_0 - \alpha_x)\alpha_0 - {\alpha_x}^2= \alpha_x(\alpha_0-\alpha_x).
\end{eqnarray*}
Suppose that $|\Po|>1$. Let $x\in \Po$ be a maximal element. Consider the natural map $\pi:\R{\alv}{\Po} \to \R{\alv_{\Po\setminus\{x\}}}{\Po \setminus \{x\}}$, which maps a tuple $(V_s)_{s\in \Po}$ to $(V_s)_{s\in \Po\setminus\{x\}}$. Map $\pi$ is surjective, as $x$ is maximal. Generic fiber of $\pi$ has the form $$\Gr(\alpha_x-X,\alpha_0-X),$$ in which $X$ is a general dimension of the sum of subspaces $V_y$ over all $y\to x$ in $\Po$, and we have
\begin{equation} \label{equalityT1}
\dim \R{\alv}{\Po} =  
\dim \Gr(\alpha_x- X,\alpha_0-X)+\dim\R{\alv_{\Po\setminus \{x\}}}{\Po \setminus \{x\}}.
\end{equation}
Applying Lemma \ref{sum} for the poset $\DD{x}$ and dimension vector 
${\alv}_{\DD{x}}$, we have that $X={\alv}_{\DD{x}}\Mi{\DD{x}}^{\,-1}{E_{\DD{x}}}$. Therefore
\begin{equation} \label{equalityT2}
	\dim \Gr(\alpha_x-X,\alpha_0-X)=(\alpha_x-{\alv}_{\DD{x}}\cdot\Mi{\DD{x}}^{\,-1}\cdot{E_{\DD{x}}})(\alpha_0-\alpha_x).
\end{equation}
By induction hypothesis we have
\begin{equation}\label{equalityT3}
\dim \R{\alv_{\Po\setminus \{x\}}}{\Po \setminus \{x\}}=\dim \mathrm{GL}(\alpha_0)-\Q{\Po\setminus\{x\}}(\alv_{\Po\setminus\{x\}}).
\end{equation}
Hence (combining \eqref{equalityT1}-\eqref{equalityT3}), 
\begin{equation*}
\dim \mathrm{GL}(\alpha_0)-\dim \R{\alv}{\Po}=\Q{\Po\setminus\{x\}}(\alv_{\Po\setminus\{x\}})-(\alpha_x-{\alv}_{\DD{x}}\cdot\Mi{\DD{x}}^{\,-1}\cdot{E_{\DD{x}}})(\alpha_0-\alpha_x).
\end{equation*}
By Lemma \ref{lem_alg} the right hand side of last equality equals $\Q{\Po}(\alv)$ and we are done. 

\section{Examples and consequences}


\begin{ex}
We will illustrate the steps in the proof of the main Theorem on the following example. Consider the poset $\Po=\{1,2,3,4,5,6,7\}$ with the order $1\prec 3, 1\prec 4, 1\prec 5, 2\prec 4, 2 \prec 5, 3\prec 6, 3\prec 7, 4\prec 6, 4 \prec 7, 5\prec 7$ and the dimension vector $\alv=(8;1,2,2,4,5,6,7)$. 
Element $6\in \Po$ is maximal with $\alpha_6=6$. Calculating $X$ as in the proof of the main Theorem we get $X=5$ (as $X$ is a general sum of subspaces with dimension $\alpha_3=2$, $\alpha_4=4$ having common subspace of dimension $1$). In this case the equality \eqref{equalityT1} has the following graphical form (where the components of dimension vectors we place at corresponding vertices of Hasse diagrams):
\[
\xymatrix@=0.5pc{
&& {8} &&&&&&& {8} 
\\ \\
& {7}  \ar@{-}[uur]  && {6} \ar@{-}[uul] &&&&&&  {7} \ar@{-}[uu]
\\ \\
{5} \ar@{-}[uur] && {4} \ar@{-}[uul] \ar@{-}[uur] && {2} \ar@{-}[uul] \ar@{-}[uulll] 
& = &  {8-5} & +  & {5}  \ar@{-}[uur]  && {4} \ar@{-}[uul]  && {2} \ar@{-}[uulll]
 \\ \\
& {2} \ar@{-}[uul] \ar@{-}[uur] && {1} \ar@{-}[uur] \ar@{-}[uul] \ar@{-}[uulll]  & & & 6-5 \ar@{-}[uu] &&& {2} \ar@{-}[uul] \ar@{-}[uur]  && {1} \ar@{-}[uur] \ar@{-}[uul] \ar@{-}[uulll]
}
\] 
The induction hypothesis yields  
$$\dim \R{\alv_{\Po\setminus \{6\}}}{\Po\setminus \{6\}}=8^2-\Q{{\Po\setminus \{6\}}}(\alv_{\Po\setminus \{6\}})=35.$$ On the other hand, $\dim \Gr(6-5,8-5)=2$. Therefore by \eqref{equalityT1} $\dim \R{\alv}{\Po}=37$ which is exactly the value of $8^2-\Q{\Po}(\alv)$.
\end{ex}
The next example shows that the assumption $\alv$ to be admissible in Theorem is necessary.
\begin{ex}
Consider the poset $\Po=\{1,2,3,4\}$ with the order $1\prec 3, 1\prec 4, 2\prec 3, 2\prec 4$ and the dimension vector $\alv=(4;2,2,3,3)$. That is, the Hasse diagram of $\Poe$ with the integers $\alpha_s, s\in \Poe$  placed at the corresponding vertices, has the following form:
\[
\xymatrix@=0.5pc{ & {4}  &&
 \\ \\
{3} \ar@{-}[uur]   && {3} \ar@{-}[uul]
 &&
 \\ \\
{2} \ar@{-}[uu] \ar@{-}[uurr]  && {2} \ar@{-}[uu] \ar@{-}[uull] &&  }
\] 
The dimension vector $\alv$ is not admissible. One checks that the dimension of of $\R{\alv}{\Po}$ is 7, while $4^2-\Q{\Po}(\alv)=8$. 

In this case we can calculate the dimension of $\R{\alv}{\Po}$ via the Euler form of enlarged poset. Namely, let $\widetilde \Po=\Po\cup\{5\}$, where $5$ is a minimal point, and let 
$\widetilde \alv=(4;1,2,2,3,3)$. 
That is, the corresponding Hasse diagram is 
\[
\xymatrix@=0.5pc{ & {4}  &&
 \\ \\
{3} \ar@{-}[uur]   && {3} \ar@{-}[uul]
 &&
 \\ \\
{2} \ar@{-}[uu] \ar@{-}[uurr]  && {2} \ar@{-}[uu] \ar@{-}[uull] && 
\\ \\ 
& {1} \ar@{-}[uur] \ar@{-}[uul] &&
}
\] 
For any point $(V_s)_{s\in \Po}\in \R{\alv}{\Po}$ the intersection between $V_1$ and $V_2$ contains a one dimensional subspace, therefore $\dim \R{\alv}{\Po}=\dim\R{\widetilde \alv}{\widetilde \Po}$. 
As $\widetilde \alv$ is an admissible dimension vector for $\widetilde \Po$, we have that $\dim\R{\widetilde \alv}{\widetilde \Po}=4^2-\Q{\widetilde \Po}(\widetilde \alv)=7$.
\end{ex}
Nevertheless there are non-admissible vector such that \eqref{eqMain} holds.
\begin{ex}
Consider the poset $\Po=\{1,2,3\}$ with the order $1\prec 3, 2\prec 3$ and the dimension vector $\alv=(4;2,2,3)$. That is, the Hasse diagram of $\Poe$ has the form:
\[
\xymatrix@=0.5pc{ & {4}  &
 \\ \\
&{3} \ar@{-}[uu]  &
 \\ \\
{2}  \ar@{-}[uur]  && {2}  \ar@{-}[uul] &&  }
\] 
The dimension vector $\alv$ is not admissible. In the contrast to the previous example, in this case 
$$\dim \R{\alv}{\Po}=7=4^2-\Q{\Po}(\alv).$$
\end{ex}



We show several consequences of equation \eqref{eqMain} for representations of posets. Recall that a \textit{subspace representation} of $\Po$ is a tuple $\V=(V_0;V_s)_{s\in \Po}$ of vector subspaces 
$V_s$ of a vector space $V_0$ such that $V_s\subseteq V_t,$ whenever $s\prec t$ in $\Po$ (for more details see \citep[Chapter 5]{simson}). The \textit{dimension vector} of $\V$ is defined as $\dimv\, \V=(\dim V_0,\dim V_s)_{s\in \Po}$. The variety $\R{\alv}{\Po}$ can be viewed as a set of all subspace representations of $\Po$ having the dimension vector $\alv$. 
The group $\mathrm{GL}(\alpha_0)$ acts on $\R{\alv}{\Po}$  via the base change so that the orbits of this action are in a bijection with the isomorphisms classes of subspace
representations of $\Po$ with the dimension $\alv$ (the reader is refered to \cite{futyus}, where the authors studied the corresponding moduli space of this action). 


Following \cite{mwz} we say that an admissible dimension vector $\alv$ is of \textit{finite type} if the group $\mathrm{GL}(\alpha_0)$ has only finitely many orbits on variety $\R{\alv}{\Po}$. We say that a non-zero $\alv' \in \A{\Po}$ is a \textit{summand} of $\alv \in \A{\Po}$ if 
$\alv-\alv'\in \A{\Po}$. It follows
from the Krull-Schmidt theorem that if $\alv$ is of finite type, then every summand of
$\alv$ is also of finite type. Theorem 2.2 from \cite{mwz} can be treated as a classification of dimensions of finite type for primitive posets (this result was generalized in \cite{drozdKub} for any poset). It trivially follows that if ${\Po}$ is a poset of finite representation type (that is, there exist only finitely many of indecomposable subspace representations of $\Po$), then any admissible dimension vector $\alv$ is of finite type. Classification of posets of finite representation type is given in \cite{kle}.  


\begin{prop}
Let $\alv$ be an admissible vector of finite type so that $Q_{\Po}(\alv)=1$, then there exists a Schurian representation
of $\Po$ with dimension vector $\alv$.
\end{prop}

\begin{proof}
Recall that a representation $\V$ is called Schurian if $\mathrm{End}(\V)$ is one dimensional. As $\alv$ is of finite type, hence the variety $\R{\alv}{\Po}$ has a dense open (in Zarisky topology) $\mathrm{GL}(\alpha_0)$ orbit $\mathcal O$. Let $\V$ be any representative from $\mathcal O$, hence 
$$
\dim \mathrm{End}(\V)=\dim \textrm{Stab}_{\mathrm{GL}(\alpha_0)}\V=\dim \mathrm{GL}(\alpha_0)-\dim \R{\alv}{\Po}=Q_{\Po}(\alv).
$$
Therefore $\V$ is Schurian.
\end{proof}

\begin{prop}
If dimension vector $\alv$ is of finite type, then 
$Q_{\Po}(\alv')\geq 1$ for any summand $\alv'$ of $\alv$.
\end{prop}
\begin{proof}
As the one-dimensional subgroup of scalar matrices in $\mathrm{GL}(\alpha_0)$ acts trivially on $\R{\alv}{\Po}$, hence
$\dim \mathrm{GL}(\alpha_0)-1\geq \dim \R{\alv}{\Po}$. Therefore $Q_{\Po}(\alv)\geq 1$. Any summand of $\alv'$ of $\alv$ is of finite type as well, therefore $Q_{\Po}(\alv')\geq 1$.
\end{proof}
The \textit{coordinate dimension vector} of $\V$ is defined as $\cdn\, \V=(c_0,c_s)_{s\in \Po}$, where $c_0=\dim V_0$, $c_s=\dim (V_s/\sum_{t\prec s} V_t)$. A representation $\V$ is said to be \textit{coordinate} if for any $s\in \Po$ the sum $\sum_{t\to s} V_t$ is direct. If $\alv\in \A{\Po}$ is of finite type, then a general representation of $\Po$ of dimension $\alv$ is coordinate. Using this fact, and taking into account the relation between $\QT{\Po}$ and $\Q{\Po}$ via the relation between $\cdn$ and $\dimv$ (see Preliminaries, and \cite[Proposition 1.1]{futyus}) we have the following analogue of Theorem 1 from \cite{drozd} formulated in terms of variety $\R{\alv}{\Po}$ and the Euler quadratic form $\Q{\Po}(\alv)$.
\newpage
\begin{prop}
Suppose that $\Q{\Po}(\alv)>0$ for any non-zero $\alv \in \A{\Po}$. Then:
\begin{itemize}
\item[(a)] the poset $\Po$ is of finite representation type;
\item[(b)] if $\Q{\Po}(\alv)=1$, then there exists unique (up to isomorphism) indecomposable subspace representation of $\Po$ having dimension $\alv$;
\item[(c)] if $\Q{\Po}(\alv)>1$, then there is no indecomposable subspace representation with dimension vector $\alv$. 
\end{itemize}
\end{prop}

\section*{Acknowledgement}
Part of this work was done during the visit of K.I. to Syracuse University in 2017; he would like to thank Mark Kleiner for the hospitality, very helpful discussions and comments. K.I. is supported in part by Fapesp grants 2014/09310-5,
2015/00116-4 and by CNPq grant 456698/2014-0.


