\section{Introduction}
\label{introduction}

In \cite{BHH}, a bilinear expansion was derived, expressing determinants of submatrices of a skew symmetric matrix 
as sums over products of  Pfaffians of their principal minors. This was shown to follow from the 
relation between the Pl\"ucker \cite{GH} and Cartan \cite{Ca} embeddings 
of Grassmannians and orthogonally isotropic Grassmannians in the projectivization of an exterior space.  
Using fermionic methods, a function theoretic realization of this identity was derived in \cite{HO}, expressing Schur functions
$s_\lambda$, labelled by integer partitions $\lambda$, as sums over products of pairs $Q_{\mu^+}Q_{\mu^-}$
of Schur Q-functions, labelled by pairs $(\mu^+, \mu^-)$ of {\em strict} partitions \cite{Mac}. Schur functions and Schur $Q$-functions  are the basic building blocks for $\tau$-functions of the KP and BKP integrable hierarchies, and the simplest case of this bilinear relation  is the well-known fact that the square of any BKP $\tau$-function is the restriction of  a KP $\tau$-function to the odd flow variables \cite{DJKM1, JM, vdLO}.

The integrable KP  and BKP hierarchies are closely related to the $A_\infty$ and  $B_\infty$  root lattices,
respectively \cite{JM}. In this work  we define, for every $\GL(\infty)$ group element $\hat{g}$, an $A_\infty$ lattice of KP $\tau$-functions,
 labelled by partitions, and for every $\SO(\infty)$ group element $\hat{h}$,  a $B_\infty$ lattice of BKP $\tau$-functions,
  labelled by strict partitions.   When $\hat{g}$ and $\hat{h}$ are suitably related (eq.~(\ref{g_h_def})), we show (Theorem \ref{KP_BKP_lattice_bilinear}) how a generalization of the bilinear expansions  of  \cite{BHH} and \cite{HO} follows,
 expressing the elements of the lattice of KP $\tau$-functions as sums over products of  pairs of  $\tau$-functions from the 
 associated lattice of BKP type.  Varying the choice of infinite orthogonal group element  
  $\hat{h}$ determining the BKP $\tau$-function at the origin of the lattice  gives all possible functional realizations of 
  such bilinear relations.  Two key ingredients underlying this result are:    the identification of {\em  polarizations} 
  (Section \ref{polarizations}) associated  with a given partition, which are just 
  pairs $(\mu^+, \mu^-)$  of strict partitions sharing the same intersections and unions,  together with the {\em factorization Lemma} \ref{factorization_lemma},   which allows us to express vacuum expectation values (VEV's) of products of mutually anticommuting
   sets of neutral fermionic operators $\{\phi^\pm_j\}_{j \in \mathbf{Z}}$ as products of VEV's of the two separate types.
  The results  of \cite{BHH} and \cite{HO} are recovered as special cases, together with other examples involving 
more general lattices of KP and BKP $\tau$-functions,  in particular, those consisting of inhomogeneous 
polynomial $\tau$-functions. 


\section{Fermionic Fock space, creation and annihilation operators, KP and BKP $\tau$-functions}
\label{fermi_ops}

\subsection{Charged fermions and KP flows}
\label{charge_fermion_KP}

We recall some definitions and notations for free fermions and KP flows \cite{JM, MJD, HB}. 
Given a separable Hilbert space ${\mathcal H}$ with orthonormal basis $\{e_j\}_{j\in \mathbf{Z}}$
and its dual space ${\mathcal H}^*$, with dual basis $\{e^j\}_{j\in \mathbf{Z}}$ satisfying
\begin{equation}
e^j(e_k) = \delta_{jk},
\label{basis_duality}
\end{equation}
the fermionic Fock space ${\mathcal F}$ is defined as the $\mathbf{Z}$-graded semi-infinite wedge product 
\begin{equation}
{\mathcal F}:= \Lambda^{\infty/2}{\mathcal H} =\sum_{n\in \mathbf{Z}} {\mathcal F}_n, 
\label{fock_def}
\end{equation}
where ${\mathcal F}_n$ denotes the $n$th  fermionic charge sector.
The elements of ${\mathcal F}$ are denoted by {\em ket} vectors $| \cdot \rangle$, while those of the dual space ${\mathcal F}^*$
are denoted by {\em bra} vectors $\langle \cdot |$. Let 
\begin{equation}
\lambda = (\lambda_1, \lambda_2, \dots, \lambda_{\ell(\lambda)}, 0, 0 \dots)
\label{part_lambda}
\end{equation}
be an extended integer partition of {\em length} $\ell(\lambda)$, consisting of a weakly decreasing
sequence $\lambda_1 \ge \cdots \ge \lambda_{\ell(\lambda})> 0$ of $\ell(\lambda)$ positive integers, 
followed by an infinite sequence  \hbox{$(\{\lambda_{\ell(\lambda)+j}=0\}_{j=1, 2, \dots}$}of $0$'s.
 An orthonormal basis $\{|\lambda; n\rangle\}$ for ${\mathcal F}_n$ is provided by the elements 
\begin{equation}
|\lambda; n\rangle\:= e_{l_1} \wedge e_{l_2} \wedge \cdots, 
\label{fock_basis}
\end{equation}
where $(l_1, l_2, \dots)$ is a strictly decreasing sequence of integers, called {\em particle positions}.
  defined in terms of the partition $\lambda$ by
  \begin{equation}
  l_j = \lambda_j -j +n, \quad j=1, 2, \dots.
  \label{part_positions}
  \end{equation}
which, after the first $\ell(\lambda)$ elements, saturate to the  sequence of all subsequent decreasing integers,
$n-\ell(\lambda), n-\ell(\lambda) -1, \dots$. The dual basis vectors are denoted $\langle \lambda; n|$, and
satisfy
\begin{equation}
\langle \lambda; n | \mu; m\rangle = \delta_{\lambda, \mu} \delta_{nm}.
\label{fock_basis_duality}
\end{equation}
The vacuum states  in each sector ${\mathcal F}_n$  correspond to the trivial partition
$\emptyset :=(0, 0, \dots)$ and are denoted
\begin{equation}
| n\rangle := |\emptyset; n\rangle.
\label{n_vac}
\end{equation}

The irreducible representation 
\begin{eqnarray}
\Gamma: {\mathcal C}_{{\mathcal H}+{\mathcal H}^*}(Q)  &\& \rightarrow \mathrm {End}({\mathcal F}) \cr
\Gamma: \xi &\& \mapsto \Gamma_\xi
\label{clifford_Gamma}
\end{eqnarray}
of the Clifford algebra ${\mathcal C}_{{\mathcal H}+{\mathcal H}^*}(Q)$ algebra 
on ${\mathcal H} + {\mathcal H}^*$ with respect to the scalar product
\begin{equation}
Q(v + \mu, w + \nu) := \nu(v) + \mu(w), \quad v,w \in {\mathcal H}, \ \mu, \nu \in {\mathcal H}^*
\label{Q_scalar_prod}
\end{equation}
 is generated by scalars and linear elements, the latter acting by exterior and interior multiplication:
 \begin{equation}
 \Gamma_v := v\wedge, \quad \Gamma_\mu := i_\mu.
 \label{clifford_rep_gen}
 \end{equation}
The fermionic creation and annihilation operator are representations of the basis elements:
\begin{equation}
\psi_j := \Gamma_{e_j} = e_j \wedge, \quad \psi^\dag := \Gamma_{e^j} := i_{e^j}
\end{equation}
and satisfy the anticommutation relations
\begin{equation}
[\psi_j, \psi_k]_+ = 0, \quad [\psi^\dag_j, \psi^\dag_k]_+ = 0, \quad [\psi_j, \psi^\dag_k]_+ = \delta_{jk}.
\label{psi_i_anticomm}
\end{equation}

For $j>0$, $\psi_{-j}$ and $\psi^\dag_{j-1}$ (resp. $\psi^\dag_{-j}$ and $\psi_{j-1}$) 
annihilate the right (resp. left) vacua:
\begin{eqnarray}
\psi_{-j} |0\rangle &\&=0, \quad \psi^\dag_{j-1} |0\rangle =0, \quad \forall  \, j >0,
\label{vac_annihil_psi_j_r}
 \\
\langle 0| \psi^\dag_{-j}  &\&=0, \quad \langle 0 | \psi_{j-1}  =0, \quad \forall  \, j >0.
\label{vac_annihil_psi_j_l}
\end{eqnarray}
The basis states may be expressed by acting with a product of pairs
of operators $\psi_j\psi^\dag_k$ on the vacuum:
\begin{equation}
|\lambda; 0\rangle =: |\lambda \rangle =  (-1)^{\sum_{j=1}^r\beta_j} \prod_{j=1}^r 
 \left( \psi_{\alpha_j} \psi_{-\beta_j-1}^\dag \right) |0\rangle ,
 \label{lambda_frob_vac}
\end{equation}
where $(\alpha_1, \dots, \alpha_r | \beta_1, \dots, \beta_r)$ are the Frobenius  indices \cite{Mac}  of the  partition $\lambda$.

The fermionic representation of elements $g\in \GL_0({\mathcal H})$ of the connected component of the 
group $\GL({\mathcal H})$ can be expressed as exponentials of $\grgl({\mathcal H})$ algebra elements, which are linear
 combinations of normal ordered products $\no{\psi_j \psi^\dag_k}$ of creation and annihilation operators \cite{JM}
\begin{equation}
\hat{g}(\tilde{A}) = e^{\sum_{i,j \in \mathbf{Z}} \tilde{A}_{jk} \no{\psi_j \psi^\dag_k}},
\label{hat_g_exp_A}
\end{equation}
where normal ordering 
\begin{equation}
\no{\hat{L}_1 \hat{L}_2} = \hat{L}_1 \hat{L}_2 - \langle 0 | \hat{L}_1 \hat{L}_2 | 0 \rangle.
\label{normal_order}
\end{equation} 
of a product of linear elements $\hat{L}_1 \hat{L}_2$ is defined so the vacuum expectation value (VEV) vanishes.
There are various conventions on restricting the permissible values of the doubly infinite matrix  elements $\tilde{A}_{jk}$,
depending on the class of solutions to the KP hierarchy one wishes to include \cite{Sa, JM, SW}.

The KP flow parameters, denoted
\begin{equation}
{\bf t}  =(t_1, t_2, \dots), 
\label{KP_flow_params}
\end{equation} 
may be interpreted as linear exponential coordinates on an infinite abelian subgroup $\Gamma_+ \subset \GL({\mathcal H})$
defined by
 \begin{eqnarray}
 \Gamma_+=\{\gamma_+({\bf t}) := e^{\sum_{j=1}^\infty t_j \Lambda^j} \}, 
 \label{Gamma_+_def}
 \end{eqnarray}
 where $\Lambda\ \in \mathrm {End}({\mathcal H})$ is the shift operator which, acts on basis elements by
 \begin{equation}
 \Lambda(e_j) = e_{j-1}.
 \label{shift_Lambda}
 \end{equation}
 The KP flows are generated by the action of $\Gamma_+$ on the infinite Grassmannian
 $\mathrm {Gr}_{{\mathcal H}_+}({\mathcal H})$ of subspaces $w\subset {\mathcal H}$ commensurate with the subspace
 \begin{equation}
 {\mathcal H}_+ = \mathrm {span}\{e_{-j}\}_{j\in \mathbf{N}^+},
 \label{HH_+_def}
 \end{equation}
 satisfying certain admissibility conditions \cite{SW, Sa}. They are mapped into the projectivization
 $\mathbf{P}({\mathcal F})$ of ${\mathcal F}$ by the Pl\"ucker map:
 \begin{equation}
 \mathfrak{P}} \def\grp{\mathfrak{p}\grl: \mathrm {span}\{w_j\}_{j\in \mathbf{N}^+} \rightarrow [w_1 \wedge  w_2 \wedge \cdots ], 
 \label{pluskcer_map}
\end{equation}
where $\{w_j\}_{j\in \mathbf{N}^+} $ is any admissible basis of $w\in\mathrm {Gr}_{{\mathcal H}_+}({\mathcal H})$.
 The image $\mathfrak{P}} \def\grp{\mathfrak{p}\grl({\mathcal H}_+)$ of ${\mathcal H}_+\subset {\mathcal H}$ is the projectivized vacuum state $[|0\rangle]$,
 and the Pl\"ucker map intertwines the action of $\Gamma_+$ induced on $\mathrm {Gr}_{{\mathcal H}_+}({\mathcal H})$
 with the (projectivized) fermionic representation of   $\Gamma_+ \subset \GL({\mathcal H})$ on ${\mathcal F}$,
 given by
\begin{equation}
\Gamma_{\gamma_+({\bf t})}=  \hat{\gamma}_+({\bf t}) := e^{\sum_{j=1}^\infty t_j J_j},
\label{hat_gamma_+}
\end{equation}
where 
\begin{equation}
J_j := \sum_{k\in \mathbf{Z}} \psi_k \psi^\dag_{k+j}, \quad j=1,2, \dots
\label{J_j_def}
\end{equation}
are the positive Fourier components of the current operator  \cite{JM}, which mutually commute
\begin{equation}
[J_n, J_m ] =0,\quad n,m \in \mathbf{N}^+.
\label{current_comp_cr}
\end{equation}
By (\ref{vac_annihil_psi_j_r}), the  current components $J_n$  annihilate the vacuum $|0\rangle$:
\begin{equation}
J_n |0 \rangle =0, \quad n\in \mathbf{N}^+
\label{J_n_vac_0}
\end{equation}
and hence $\Gamma_+$ stabilizes the vacuum
\begin{equation}
\hat{\gamma}_+({\bf t}) |0\rangle = |0\rangle.
\label{gamma_+_vac}
\end{equation}

For each $g\in \GL_0({\mathcal H})$, we define the corresponding KP $\tau$-function
\begin{equation}
\tau^{KP}_g({\bf t}) = \langle 0 | \hat{\gamma}_+({\bf t})\hat{g} | 0 \rangle ,
\label{KP_tau_fermi}
\end{equation}
which satisfies the Hirota bilinear residue equations \cite{Sa, JM} of the KP hierarchy.
More generally, $\hat{g}$ need not be a  $\GL_0({\mathcal H})$ group element,
 but any element of the Clifford algebra\index{Clifford algebra}  satisfying the 
 bilinear commutation relation
\begin{equation}
\left[ {\mathcal I} ,  \hat{g} \otimes \hat{g} \right] = 0, 
\label{bilinear_commut}
\end{equation}
where ${\mathcal I}$ is the endomorphism of  ${\mathcal F} \otimes {\mathcal F}$ defined by
\begin{equation}
{\mathcal I}:= \sum_{j\in \mathbf{Z}} \psi_j \otimes \psi_j^\dag.
\label{I_bilinear}
\end{equation}

\subsection{Neutral fermions and BKP flows}
\label{neutral_fermionc_BKP}

 Neutral fermions $\phi^+_j$ and $\phi^-_j$ are defined \cite{DJKM1, DJKM2, JM, You, KvdL1} by 
\begin{equation}
\phi^+_j :=\frac{\psi_j+ (-1)^j\psi^\dag_{-j}}{\sqrt 2},\quad 
\phi^-_j :=i\frac{\psi_j-(-1)^j \psi^\dag_{-j}}{\sqrt 2},\quad j \in \mathbf{Z}
\label{neutral_charged}
\end{equation}
(where $i=\sqrt{-1}$), and satisfy
\begin{equation}
 [\phi^+_j,\phi^-_k]_+=0,\quad [\phi^+_j,\phi^+_k]_+ = [\phi^-_j,\phi^-_k]_+ =(-1)^j \delta_{j+k,0}.
 \label{neutral_ccr}
\end{equation}
In particular,
\begin{equation}
(\phi^+_0)^2=(\phi^-_0)^2=\tfrac{1}{2}.
\label{square_phi_0}
\end{equation}
Acting on the vacua $|0\rangle$ and $|1 \rangle$, we have
\begin{eqnarray}
\phi^+_{-j}|0\rangle &\&= \phi^-_{-j} |0\rangle =\phi^+_{-j}|1\rangle = \phi^-_{-j} |1\rangle =0 , \quad  \forall  j > 0  , \quad \forall  j > 0, 
\label{phi_pm_j_01vac_r} \\
\langle 0| \phi^+_{j} &\&= \langle 0|\phi^-_{j}=  \langle 1| \phi^+_{j} = \langle 1|\phi^-_{j}  =0 , \quad  \forall  j > 0 ,
\label{phi_pm_j_01vac_l}
\\
\phi^+_0|0\rangle &\& = 
- i \phi^-_0 |0\rangle =
\tfrac{1}{\sqrt{2}} \psi_0|0\rangle = \tfrac{1}{\sqrt{2}} |1\rangle  ,
\label{phi_pm_0vac_r}    
\\
 \langle 0| \phi^+_0 &\& = 
 i \langle 0|\phi^-_0  =
\tfrac{1}{\sqrt{2}} \langle  0 | \psi_0^\dag = \tfrac{1}{\sqrt{2}}\langle 1|.
\label{phi_pm_0vac_l}  
\end{eqnarray}

Let $\tilde{{\mathcal H}}^\pm \subset {\mathcal H} + {\mathcal H}^*$ be the two mutually orthogonal subspaces  defined
by
\begin{eqnarray}
\tilde{{\mathcal H}}^+  &\&:= \mathrm {span}\{f_j^+:=\frac{1}{\sqrt{2}} ( e_j + (-1)^{-j} e^{-j}\}_{j\in \mathbf{Z}}, \cr
 \cr
\tilde{{\mathcal H}}^-&\& := \mathrm {span}\{f_j^-:= \frac{i}{\sqrt{2}} ( e_j - (-1)^{-j} e^{-j}\}_{j\in \mathbf{Z}} .
\end{eqnarray}
Restricting  the scalar  product $Q$ to each of these subspaces, we have the induced scalar products $Q_\pm$ on ${\mathcal H}^\pm$
defined by
\begin{eqnarray}
Q_+(f^+_j, f^+_k) &\&:=  Q(f^+_j, f^+_k) = (-1)^j\delta_{j+k, 0}, \cr
&\& \cr
 Q_-(f^-_j, f^-_k) &\&:= Q(f^-_j, f^-_k) = (-1)^j\delta_{j+k, 0}. 
 \label{Q_pm_def}
\end{eqnarray}
We thus have two mutually commuting fermionic representations of the orthogonal algebras $\gro(\tilde{{\mathcal H}}^\pm,Q_\pm)$
defined by the span of the bilinear elements $\{\phi^\pm_j\phi^\pm_k\}_{j,k \in \mathbf{Z}}$. 

The fermionic representations of elements $h^\pm \in \SO(\tilde{{\mathcal H}}^\pm, Q_\pm)$ in the connected component of the corresponding 
mutually commuting orthogonal  groups $\SO(\tilde{{\mathcal H}}^\pm, Q_\pm)$, are defined by exponentiation \cite{DJKM1, DJKM2, vdLO}
\begin{equation}
\hat{h}^\pm(A) := e^{\tfrac{1}{2}\sum_{j,k \in \mathbf{Z}}  A_{jk} \no{\phi^\pm_j \phi^\pm_{k}} },
\label{hat_h_alpha_pm_exp_A}
\end{equation}
 where $\{A_{jk}\}_{j,k \in \mathbf{Z}}$ are the elements of a doubly infinite skew symmetric matrix $A$.
Let ${\mathcal F}_{\phi^\pm}\subset {\mathcal F}$ be the two subspaces of ${\mathcal F}$ defined as
\begin{equation}
{\mathcal F}_ {\phi^+}:= \mathrm {span}\{|\hat{\alpha}^+ \big)\}, \quad 
{\mathcal F}_ {\phi^-}:= \mathrm {span}\{|\hat{\alpha}^- \big)\}
 \label{FF_pm_def}
\end{equation}
where
\begin{equation}
|\alpha^+ \big) := \phi^+_{\alpha_1} \cdots \phi^+_{\alpha_r}|0 \rangle, \quad
|\alpha^- \big) := \phi^-_{\alpha_1} \cdots \phi^-_{\alpha_r}|0 \rangle.
 \label{alpha_pm}
\end{equation}
Then ${\mathcal F}_ {\phi^+}$ and ${\mathcal F}_ {\phi^-}$ are invariant under the fermionic representations
(\ref{hat_h_alpha_pm_exp_A}) of the groups $\SO(\tilde{{\mathcal H}}^\pm, Q_\pm)$.

The positive neutral fermion current components $J^{B+}_j$ and $\hat{J}^{B-}_j$ are defined as
\begin{equation}
J^{B+}_j:=\tfrac 12 \sum_{k\in\mathbb{Z}} (-1)^{k+1}\phi^+_k \phi^+_{-k-j},\quad 
J^{B-}_j=\tfrac 12 \sum_{k\in\mathbb{Z}} (-1)^{k+1}\phi^-_k\phi^-_{-k-j}, \quad j\in \mathbf{N}^+,
\label{J_j_B_def}
\end{equation}
from which it follows that the even components $\{J^{B\pm}_{2j}\}$ vanish, while the odd ones mutually commute,
\begin{equation}
[J^{B+}_{2j-1}, J^{B+}_{2k-1}]  = 0,\quad [J^{B-}_{2j-1}, J^{B-}_{2k-1}]=0,\quad
[J^{B+}_{2j-1}, J^{B-}_{2k-1}] =0, \quad j,k \in \mathbf{N}^+ .
\label{JB_pm_crs}
\end{equation}

It also follows, from  (\ref{phi_pm_j_01vac_r}), that the neutral current components $J^{B\pm}_j$  
 annihilate the vacua $|0\rangle$ and $|1\rangle$,
\begin{equation}
J^{B+}_j|0 \rangle = 0, \quad J^{B-}_j|0 \rangle =0, \quad J^{B+}_j|1 \rangle = 0, \quad J^{B-}_j|1 \rangle, \quad j \in \mathbf{N}^+, 
\label{J_B_vac}
\end{equation}
and, from (\ref{J_j_def}),  (\ref{J_j_B_def}),  that:
\begin{lemma}\em
For  odd $n= 2j-1$, we have
\begin{equation}
 J_{2p-1}=J^{B+}_{2j-1}+J^{B-}_{2j-1}, \quad j \in \mathbf{N}^+.
 \label{J_J_Bpm}
\end{equation}
\end{lemma}

The  BKP flow parameters are denoted
\begin{equation}
{\bf t}_B:=(t_1, t_3, \dots ),
\label{BKP_flow_params}
\end{equation}
and these may be viewed as determining a subset  $\{\mathbf{t}'\} $ of the $\mathbf{t}$'s , where
\begin{equation}
\mathbf{t}':=(t_1,0,t_3,0,t_5,\dots).
\label{t_prime_def}
\end{equation}
Following \cite{DJKM1, DJKM2,  You, JM, vdLO}, we  define  two mutually commuting abelian groups of BKP flows 
\hbox{$\Gamma^{B+} = \{\gamma^{B+}({\mathbf{t}}_{B+})\}$} and 
\hbox{$\Gamma^{B-} = \{\gamma^{B-}({\mathbf{t}}_B)\}$}, 
with Clifford representations
\begin{equation}
\hat{\gamma}^{B+}({\mathbf{t}}_B) :=  e^{\sum_{j=0}^\infty J^{B+}_{2j-1} t_{2j-1} } ,
 \quad \hat{\gamma}^{B-}({\mathbf{t}}_B) := e^{\sum_{j=1}^\infty J^{B-}_{2j-1} t_{2j-1} }.
 \label{hat_gamma_B_pm}
\end{equation}
By (\ref{J_B_vac}), these  both stabilize the vacua $|0\rangle$ and $| 1\rangle$
\begin{eqnarray}
\hat{\gamma}^{B+}({\mathbf{t}}_B)|0\rangle &\&= |0\rangle, \quad \hat{\gamma}^{B+}({\mathbf{t}}_B)|1\rangle = |1\rangle, \cr
\hat{\gamma}^{B-}({\mathbf{t}}_B)|0\rangle &\&= |0\rangle, \quad \hat{\gamma}^{B-}({\mathbf{t}}_B)|1\rangle = |1\rangle.
\label{J_sum_J+-}
\end{eqnarray}
It also follows that
\begin{equation}
\hat{\gamma}_+({\bf t}') = \hat{\gamma}^{B+}_+({\bf t}_B)  \hat{\gamma}^{B-}_+({\bf t}_B) .
\label{gamma_factoriz}
\end{equation}
The BKP $\tau$-function, defined as
\begin{equation}
\tau^{BKP}_h({\bf t}_B) = \langle 0| \hat{\gamma}^{B+}({\bf t}_B) \hat{h}^+ | 0 \rangle = \langle 0| \hat{\gamma}^{B-}({\bf t}_B) \hat{h}^- | 0 \rangle,
\label{BKP_tau_fermi}
\end{equation}
then satisfies the BKP Hirota bilinear residue equations \cite{DJKM1, DJKM2, JM}.

More generally, $\hat{h^\pm}$ need not be $\SO(\tilde{{\mathcal H}}^\pm)$ group elements for
(\ref{BKP_tau_fermi}) to define a BKP $\tau$-function.
Define ${\mathcal I}_B^\pm\in \mathrm {End}({\mathcal F}_{\phi^\pm} \otimes {\mathcal F}_{\phi^\pm})$ as
\begin{equation}
{\mathcal I}_B^\pm := \sum_{j\in \mathbf{Z}}(-1)^j \phi^\pm_j  \otimes \phi^\pm_{-j} ,
\label{II_B}
\end{equation}
and let $P^\pm \in  \mathrm {End}({\mathcal F}_{\phi}^{\pm})$ be  defined on the bases $\{|\alpha^\pm)\}$ as
\begin{equation}
P^{\pm}|\alpha^\pm) := (-1)^{m(\alpha^\pm)}|\alpha^\pm), 
\label{FF_partity_op}
\end{equation}
where $m(\alpha^\pm)$ is the length (i.e.~the cardinality) of $\alpha^\pm$.
Then  \ref{BKP_tau_fermi}) defines a BKP $\tau$-function \cite{HB, DJKM1, DJKM2, JM} provided the bilinear relations 
  \begin{equation}
{\mathcal I}_B^\pm  | v^\pm) \otimes |v^\pm ) =   (\phi_0 \otimes \phi_0)P^\pm | v^\pm ) \otimes  P^\pm| v^\pm ).
\label{neutral_fermion_bilinear}
\end{equation}
are satisfied by the elements 
\begin{equation}
|v^\pm) = \hat{h}^\pm|0\rangle.
\label{decomp_FF_N_pm)el}
\end{equation}

We also have the following key factorization lemma \cite{HO, You, JM}.
 \begin{lemma}[\bf Factorization]
 \label{factorization_lemma}
 If $U^+$ and $U^-$ are either even or odd degree elements of the subalgebra generated by the operators
 $\{\phi^+_i\}_{i \in \mathbf{Z}}$ and  $\{\phi^-_i\}_{i \in \mathbf{Z}}$ respectively, the VEV of their product can be factorized as:
\begin{equation}
\langle 0 | U^+ U^-|0\rangle =
\begin{cases}
\langle 0 |  U^+|0\rangle \langle 0| U^- |0\rangle 
&  
\text{ if } U^+ \text{ and } U^-\text{ are both of even degree}\\
0  &  \text{ if } U^+ \text{ and }  U^- \text{ have different  parity }\\ 
2i \langle 0|U^+\phi^+_0|0\rangle \langle 0|U^-\phi^-_0 |0\rangle &  
\text{  if }  U^+ \text{ and }  U^+ \text{ are both of odd degree}.
\end{cases}
\label{gen_factorization}
\end{equation}
\end{lemma} 

Defining
\begin{equation}
g(h(A)) = h^+(A)h^-(A), \quad \hat{g}(h(A)) :=\hat{h}^+(A)\hat{h}^-(A)
\label{g_h_def}
\end{equation}
and using the relations (\ref{neutral_charged}) between neutral and charged fermions and the mutual commutativity
of the algebras $\gro(\tilde{{\mathcal H}}^\pm,  Q_\pm)$, we can re-express $\hat{g}$ as the fermionic representation
of an element of $\GL({\mathcal H})$
\begin{equation}
\hat{g}(\tilde{A})=\hat{g}(h(A)) = e^{\sum_{j,k \in \mathbf{Z}} (-1)^kA_{j, -k} \no{\psi_j \psi^\dag_k}}
\label{fermi_rep_g_h}
\end{equation}
where
\begin{equation}
\tilde{A} = (-1)^k A_{j, -k}
\label{tilde_A_A_skew}
\end{equation}
 Comparing formulae (\ref{KP_tau_fermi}) and  (\ref{BKP_tau_fermi}) for the KP $\tau$-function
 $\tau^{KP}_{g(h)}({\bf t}')$ and the BKP $\tau$-function  $\tau^{BKP}_h({\bf t}_B)$, using
(\ref{gamma_factoriz}) and (\ref{g_h_def}) and Lemma \ref{factorization_lemma}, we obtain the
following well-known result \cite{DJKM1,  DJKM2}
\begin{equation}
\tau^{KP}_{g(h)}({\bf t}') = \left(\tau^{BKP}_h({\bf t}_B) \right)^2.
\label{tau_KP_g_h}
\end{equation}

The next section is aimed at obtaining the general bilinear relation expressing  a lattice of KP $\tau$-functions,  
labelled by partitions, associated with  $\tau^{KP}_{g(h)}({\bf t}')$,  as sums over products of the elements  
of a lattice of BKP $\tau$-functions, lablled by strict partitions, similarly associated with $\tau^{BKP}_h({\bf t}_B)$

\section{Bilinear expansions of KP $\tau$-function lattices in BKP $\tau$-functions}
\label{ bilnear_KP_BKP}
In this section, we define lattices of KP and BKP $\tau$-functions related to a given pair 
\hbox{$\tau^{KP}_{g(h)}(\mathbf{t})$, $\tau^{BKP}_h({\bf t}_B)$}, labelled by partitions and strict partitions, 
respectively, and derive a bilinear expansion for the first of these in terms of the second.


\subsection{Lattices of KP and BKP $\tau$-functions}
\label{KP_BKP_tau_lattice}
In the following, we view the KP and BKP flow variables ${\bf t}$ and ${\bf t}_B$, 
which are the arguments of our $\tau$-functions, as independent coordinates.
To make contact with the usual definitions of Schur functions and Schur $Q$-functions,
as symmetric polynomials, or symmetric functions of a finite or infinite set of variable $\{x_j\}_{j\in\mathbf{N}^+}$,
we recall that the variables ${\bf t}$ and ${\bf t}_B$ may be restricted to equal the (normalized) power sums
in these, 
\begin{equation}
t_i = {1\over i} \sum_{j=1}^N{x_j^i},
\end{equation}
(where N could be a finite positive integer or $\infty$).
We then denote the infinite sequence ${\bf t}=(t_1, t_2 \dots )$ as
\begin{equation}
{\bf t}=  [{\bf x}] := ([{\bf x}]_1, [{\bf x}]_2, \dots), \quad [{\bf x}]_i :=   {1\over i} \sum_{j=1}^N{x_j^i}, \quad i \in \mathbf{N}^+.
\label{t_norm_power_sum}
\end{equation}
However, when we specialize to the values $\mathbf{t} = \mathbf{t}'$ , in which all the even flow variables vanish,
we cannot interpret these as the usual power sums (\ref{t_norm_power_sum}). Instead, in order to
restrict these to symmetric polynomials, we use the {\em supersymmetric} parametrization \cite{Iv}
\begin{equation}
\mathbf{t}' = [{\bf y}] - [-{\bf y}], \quad t_{2i-1} = {2\over 2i-1} \sum_{j =1}^Ny_j^{2i-1},
\label{t_prime_y_sum}
\end{equation}
where ${\bf y} =(y_1, y_2, \dots)$ is an auxiliary (finite or infinite) sequence of independent variables.

For any KP $\tau$-function $\tau^{KP}_g(\mathbf{t})$, we associate an $A_\infty$ lattice of KP $\tau$-functions
\begin{equation}
\pi_{(\alpha|\beta)}(g) ({\bf t}):= \langle 0 | \hat{\gamma}_+({\bf t}) \hat{g} | \lambda \rangle,
\label{pi_alpha_beta}
\end{equation}
labelled by partitions $\lambda =(\alpha|\beta)$  and, for any   BKP $\tau$-function $\tau^{BKP}_h(\mathbf{t}_B)$,
a $B_\infty$ lattice of BKP $\tau$-functions
\begin{equation}
\kappa_{\alpha} (h)({\bf t}_B) := \langle 0 | \hat{\gamma}^{B+}({\bf t}_B) \hat{h}^+ | \alpha^+ \big) =
 \langle 0 | \hat{\gamma}^{B-}({\bf t}_B) \hat{h}^- | \alpha^- \big),
 \label{kappa_alpha}
\end{equation}
labelled by strict partitions $\alpha$ of even cardinality $r$.
\begin{remark}
Note that if we change the elements $\hat{g} \rightarrow \hat{g} \hat{g}_0 $ and 
$\hat{h}^\pm \rightarrow \hat{h} \hat{h}_0 $ in (\ref{pi_alpha_beta}) and (\ref{kappa_alpha})  by right multiplication by elements 
$\hat{g}_0$,  $\hat{h}_0$ of the stabilizer of the vacuum $|0\rangle$, this has no effect
on the $\tau$-functions $\tau^{KP}_g(\mathbf{t})$ and $\tau^{BKP}_h(\mathbf{t}_B)$, but
will change the other elements $\pi_{(\alpha|\beta)}(g)(\mathbf{t})$, $\kappa_{\alpha} (h)({\bf t}_B) $
of the associated lattices.
 

\end{remark}

By Wick's theorem, $\pi_{(\alpha|\beta)}(g) ({\bf t})$ is the determinant of the matrix 
$\pi_{(\alpha_j|\beta_k)}(g)({\bf t}) $ corresponding to hook partitions $(\alpha_j|\beta_k)$ 
\begin{equation}
\pi_{(\alpha|\beta)}(g)({\bf t}) = \mathrm {det}\left(\pi_{(\alpha_j|\beta_k)}(g)({\bf t}) \right)_{1\le j, k\le r}, 
\label{pi_alpha_beta_det}
\end{equation}
and  $\kappa_{\alpha} (h)({\bf t}_B)$ is the Pfaffian of the skew matrix $\kappa_{(\alpha_j, \alpha_k)} (h)({\bf t}_B)$
corresponding to the Frobenius rank $r=2$ strict partitions $\alpha = (\alpha_j, \alpha_k)$:
\begin{equation}
\kappa_{\alpha} (h)({\bf t}_B)  = \mathrm {Pf}\left( \kappa_{(\alpha_j, \alpha_k)} (h)({\bf t}_B)\right)_{1\le j, k\le r}.
\label{kappa_alpha_pfaff}
\end{equation}
where, for $\alpha_k\ge\alpha_j$, $\kappa_{(\alpha_j, \alpha_k)(h) ({\bf t}_B)} $ is  defined  by
\begin{equation}
\kappa_{(\alpha_j, \alpha_k)}(h)({\bf t}_B)  := - \kappa_{(\alpha_k, \alpha_j)}(h)({\bf t}_B) .
\label{kappa_alpha_def}
\end{equation}



\subsection{Polarizations}
\label{polarizations}

\begin{definition}
A {\em polarization} of $(\alpha|\beta)$,  is a pair  $(\mu^+, \mu^-)$
of strict partitions with cardinalities  (or {\em lengths})
\begin{equation}
m(\mu^+):= \#(\mu^+), \quad  m(\mu^-)):= \#(\mu^-)
\label{m_pm_def}
\end{equation}
(including  possibly a zero part $\mu^+_{m(\mu^+)}=0$ or $\mu^-_{m(\mu^-)}=0$), satisfying
\begin{equation}
\mu^+ \cap \mu^- = \alpha \cap I(\beta), \quad \mu^+ \cup \mu^- = \alpha \cup I(\beta),
\label{mu_pm_cap_cup}
\end{equation}
where
\begin{equation}
I(\beta) := (I_1(\beta), \dots I_r(\beta))
\label{I_beta}
\end{equation}
is the strict partition \cite{Mac} with parts
\begin{equation}
I_j(\beta) = \beta_j +1, \quad j=1, \dots r.
\label{I_j_beta}
\end{equation}
The set of all polarizations of $(\alpha|\beta)$  is denoted ${\mathcal P}(\alpha,\beta)$.
\end{definition}
We denote the strict partition obtained by intersecting  $\alpha$ with $I(\beta)$ as
\begin{equation}
S := \alpha \cap I(\beta),
\label{S_def}
\end{equation}
and its cardinality as
\begin{equation}
s := \#(S).
\end{equation}
Since both $\alpha$ and $I(\beta)$ have cardinality $r$, it follows that
\begin{equation}
m(\mu^+) + m(\mu^-) = 2r,
\label{m_pm_sum}
\end{equation}
so $m(\mu^\pm)$ must have the same parity.   It is easily verified \cite{HO} that the cardinality of ${\mathcal P}(\alpha, \beta)$ is $2^{2r -2s}$.
The following was proved in \cite{HO}.
\begin{lemma}[\bf Binary sequence associated to a polarization]
For every polarization $\mu:=(\mu^+, \mu^-)$ of $\lambda=(\alpha| \beta)$ there is a unique binary sequence
of length $2r$
\begin{equation} 
\epsilon(\mu) =(\epsilon_1(\mu), \dots, \epsilon_{2r}(\mu)),
\label{epsilon_mu_binary}
\end{equation}
with
\begin{equation}
\epsilon_j(\mu) =\pm, \quad j=1, \dots 2r,
\label{epsilon_j_binary}
\end{equation}
such that 
\begin{enumerate}
\item The sequence of pairs
\begin{equation}
((\alpha_1, \epsilon_1(\mu)), \cdots  (\alpha_r, \epsilon_r(\mu)), (\beta_1+1, \epsilon_{r+1}(\mu)), \dots , ( \beta_r+1, \epsilon_{2r}(\mu))
\label{alpha_beta_sequence}
\end{equation}
is a permutation of the sequence
\begin{equation}
((\mu^+_1,+), \cdots ( \mu^+_{m^+(\mu)},+), (\mu^-_1, -),  \dots  , (\mu^-_{m^-(\mu)}, -)
\label{mu_sequence}
\end{equation}
\item
\begin{equation}
 \epsilon_j(\mu)= + \ \text{ if } \alpha_j \in S, \ \text{ and } \epsilon_{r+j}(\mu)= - \ \text{ if } \beta_j +1\in S, \quad j=1, \dots, r.
 \label{canon_epsilon_constraint}
 \end{equation}
\end{enumerate}
\end{lemma}

\begin{definition}
 The {\em sign} of the polarization $(\mu^+, \mu^-)$, denoted $\sgn(\mu)$, is defined 
as the sign of the permutation  that takes the sequence (\ref{alpha_beta_sequence})  into the sequence
(\ref{mu_sequence}).
\end{definition}
Denote by
\begin{equation}
\pi(\mu^\pm) :=\#(\alpha \cap \mu^{\pm})
\label{pi_pm_mu_def}
\end{equation}
the cardinality of the intersection of $\alpha$ with $\mu^{\pm}$. It follows that
\begin{equation}
\pi(\mu^+) + \pi(\mu^-) = r+s. 
\label{pi_pm_sum}
\end{equation}
 We then have
 \begin{lemma}
\begin{equation}
|\lambda \rangle =  {(-1)^{\tfrac{1}{2}r(r+1) + s}\over 2^{r-s}} \sum_{\mu\in {\mathcal P}(\alpha, \beta)}
\sgn(\mu)(-1)^{\pi(\mu^-)} i^{ m(\mu^-)}\prod_{j=1}^{m(\mu^+)} \phi^+_{\mu^+_j} \prod_{k=1}^{m(\mu^-)} \phi^-_{\mu^-_k}  |0 \rangle.
\label{polariz_sum}
\end{equation}
\end{lemma}
\begin{proof}
 In  eq.~(\ref{lambda_frob_vac}), reorder the product over the factors $\psi_{\alpha_j} \psi^\dag_{-\beta_j-1}$,   so 
 the  $\psi_{\alpha_j}$ terms precede the $\psi^\dag_{-\beta_j-1}$ ones, giving an overall sign 
 factor $(-1)^{\tfrac{1}{2}r(r-1)}$.  Then substitute 
 \begin{equation}
\psi_{\alpha_j} = \tfrac{1} {\sqrt{2}}( \phi^+_{\alpha_j} - i \phi^-_{\alpha_j}), \quad
 \psi^\dag_{-\beta_j-1}= \tfrac{(-1)^j}{ \sqrt{2}} (\phi^+_{\beta_j+1} + i \phi^-_{\beta_j+1}), \quad j\in \mathbf{Z},
\end{equation}
which follows from (\ref{neutral_charged}),
for all the factors, and expand  the product as a sum over monomial terms of the form
\begin{equation}
 \prod_{j=1}^{m(\mu^+)} \phi^+_{\mu^+_j} \prod_{k=1}^{m(\mu^-)} \phi^-_{\mu^-_k}  |0 \rangle.
 \end{equation}
 Taking into account the sign factor $\sgn(\mu)$ corresponding to the order of the neutral fermion
factors, as well as the powers of $-1$ and $i$, and noting that there are $2^s$  resulting identical terms, 
then gives (\ref{polariz_sum}).
\end{proof}

\begin{definition}{\bf Supplemented partitions.}
If $\nu$ is a strict partition of cardinality $m(\nu)$ (with $0$ allowed as a part), 
we define the associated {\em supplemented partition} $\hat{\nu}$ to be
\begin{equation}
\hat{\nu} := \begin{cases}  \nu \ \text{ if } m(\nu) \ \text{ is even}, \cr
                  (\nu,0)  \  \text{ if } m(\nu) \  \text{ is odd}.
                  \end{cases}
 \label{hat_nu}
\end{equation}
We denote by $m(\hat{\nu})$  the cardinality of $\hat{\nu}$.
\end{definition}

\subsection{Bilinear expansion theorem}
\label{KP_BKP_bilinear}

Combining the above, we arrive at our main  result. 
\begin{theorem}
\label{KP_BKP_lattice_bilinear}
Restricting the group element $\hat{g}$ to $\hat{g}(h)=\hat{h}^+ \hat{h}^-$ and the $\mathbf{t}$ values to $\mathbf{t}'$, 
the lattice of KP $\tau$-functions $\pi_{(\alpha|\beta)}(g(h))({\bf t}')$ may be expressed as sums over products
of pairs of BKP $\tau$-functions $\kappa_{\hat{\mu}^+} (h)({\bf t}_B)$ and $\kappa_{\hat{\mu}^-} (h)({\bf t}_B)$ as follows
\begin{equation}
\pi_{(\alpha|\beta)}(g(h))({\bf t}')= {(-1)^{\tfrac{1}{2}r(r+1) + s}\over 2^{r-s}}\sum_{\mu\in {\mathcal P}(\alpha, \beta)}
\sgn(\mu)(-1)^{\pi(\mu^-) +\tfrac{1}{2}m(\hat{\mu}^-)}\kappa_{\hat{\mu}^+} (h)({\bf t}_B)\kappa_{\hat{\mu}^-} (h)({\bf t}_B).
\label{bilinear_exp}
\end{equation}
\end{theorem}
\begin{proof}
Substituting (\ref{polariz_sum}) into (\ref{pi_alpha_beta}), using (\ref{kappa_alpha}) and (\ref{alpha_pm}) and
applying the factorizations (\ref{g_h_def}), (\ref{gamma_factoriz}) and Lemma \ref{factorization_lemma}
 gives the bilinear expansion \ref{bilinear_exp}.
\end{proof}

\section{Examples}
\label{special_cases}
The first two examples, \ref{det_pfaff_bilin_sum} and \ref{schur_Q_schur_bilin_sum},  show how the results 
of \cite{BHH} and \cite{HO} are recovered by choosing special values for the parameters in Theorem \ref{KP_BKP_lattice_bilinear} . 
Examples (\ref{symm_polynom_bilin_sum})  and (\ref{r_p_polynom_bilin_sum}) give families of lattices of  
 polynomial  $\tau$-functions of KP and BKP type that generalize the 
Schur functions and Schur $Q$-functions \cite{BL, HL, KvdL1, KvdL2, KvdLRoz, Iv}. 
These include, as special cases: {\em factorial}, {\em shifted}  and {\em interpolation Schur functions}, \cite{BL, OkOl, Ol} 
{\em interpolation} Schur $Q$-functions \cite{Iv}, and multivariate Laguerre polynomials \cite{La, Ol}.


\begin{example}[\bf Determinants of skew matrices as bilinear sums over Pfaffians]
\label{det_pfaff_bilin_sum}

The bilinear identity in \cite{BHH} follows as a particular case of Theorem \ref{KP_BKP_lattice_bilinear},
 Setting \hbox{${\bf t} = {\bf 0} = (0, 0, \dots)$} in (\ref{bilinear_exp}),
  the $\tau$-function $\pi_{\alpha |\beta)}({\bf 0})(g)$ becomes the  $(\alpha|\beta)$ Pl\"ucker coordinate
of the element $g^T({\mathcal H})$ of the Grassmannian $\mathrm {Gr}_{{\mathcal H}}({\mathcal H}+{\mathcal H}^*)$ of subspaces of ${\mathcal H}+{\mathcal H}^*$ commensurate
with the subspace ${\mathcal H}$ which is mapped, under the Pl\"ucker map, to the vacuum $|0\rangle$.

In the big cell of the Grassmannian, which consists of those elements that are the graph of a map $w: {\mathcal H}\rightarrow {\mathcal H}^*$,
it is the determinant of minors of the  affine coordinate matrix $A$ which, for $g$ of the form
(\ref{g_h_def}) is skew symmetric. And $\kappa_{\hat{\mu}^\pm} (h)({\bf 0}_B)$ are the {\em Cartan coordinates} \cite{BHH} of
the element $h^T({\mathcal H})$ of the maximal  isotropic Grassmannian $\mathrm {Gr}^0_{\mathcal H}({\mathcal H}+{\mathcal H}^*, Q)$, which is the
Pfaffian of the principal minors of the affine coordinate matrix $A$ corresponding to the strict partitions $\mu^\pm$.
This reproduces the expansions of the determinants of minors of a skew matrix as sums over products of Pfaffians of their
principal minors given by Corollary 6.4 of \cite{BHH}. 
\begin{equation}
 \mathrm {det}(A_{(I | J)}) = {(-1)^{\frac{1}{2}r(r+1)+s}\over 2^{r-s} } {\hskip -10 pt}\sum_{(K,L) \in {\mathcal P}(I, \beta)}
   {\hskip -10 pt} (-1)^{\pi +l/2} \sgn(K,L) \mathrm {Pf}(A_{(K | K)})\mathrm {Pf}( A_{(L | L)}), 
 \label{single_sum_pfaff_CB_ident}
 \end{equation}
 where $A$ is a skew symmetric $N \times N$ matrix, $I$ and $J$ are the rows and columns
 \begin{equation}
 I=(I_1, \dots, I_r) =I(\alpha) , \quad J=(J_1, \dots, I_r)=I(\beta)
 \label{I_J_def}
 \end{equation}
of the $r \times r$ minor $A_{(I | J)}$, 
\begin{equation}
K=(K_1, \dots, K_k)= I(\mu^+), \quad L=(L_1, \dots, L_l)= I(\mu^-)
\label{K_L_def}
\end{equation}
are the rows and columns of the pair of principal minors $A_{(K| K)}$ and $A_{(L | L)}$, and
\begin{equation}
\pi:=\#(I\cap L),\quad k=m(\mu^+)=\#(K), \quad  l=m(\mu^-)=\#(L), 
\label{k_l_def_sum}
\end{equation}
The ordered subsets $I,J, K, L \subset (1, \dots N)$ are polarizations related by
\begin{equation}
I\cap J = K\cap L:=S, \quad I\cup J = K\cup L.
\label{I_cap_cupJ}
\end{equation}
$s$  is the cardinality of $S$, and both $k$ and $l$ must have even parity
 in order that the Pfaffians in the sum not vanish.

\end{example}


 \begin{example}[\bf Schur functions as bilinear sums over Schur $Q$-functions]
\label{schur_Q_schur_bilin_sum}

If we set $h$ equal to the identity element $\mathbf{I}\in \SO({\mathcal H}+{\mathcal H}^*, Q)$, the  $\tau$-function $\pi_{\alpha|\beta)}({\bf t}')(\mathbf{I})$
becomes the Schur function  \cite{DJKM1, DJKM2, JM, HB}
\begin{equation}
s_{(\alpha|\beta)}({\bf t}) =  \pi_{\alpha|\beta)}(\mathbf{I})({\bf t}) =\langle 0 | \hat{\gamma}_+({\bf t})| \lambda \rangle, 
\label{schur_fermionic}
\end{equation}
which, by the Giambelli identity (which is a particular case of (\ref{pi_alpha_beta_det})), is the determinant 
of the matrix formed from Schur functions corresponding to hook partitions, whose restriction 
to ${\bf t}'$ equals the skew matrix formed from the Schur $Q$-functions $Q_{jk}({\bf t}_B)$ with  Frobenius rank $2$.
 Then $\kappa_{\hat{\mu}^\pm} (\mathbf{I}_{{\mathcal H}^\pm_0})({\bf t}_B))$ become the  (scaled) Schur $Q$-functions 
 \begin{eqnarray}
{\mathcal Q}_{\hat{\mu}^\pm}({\bf t}_B)&\&= \kappa_{\hat{\mu}^\pm} (\mathbf{I}_{{\mathcal H}^\pm_0})({\bf t}_B))
= \langle 0 |  \hat{\gamma}^{B\pm} ({\bf t}_B) | \hat{\mu}^\pm) \cr
&\&= 2^{-\frac{1}{2}m(\hat{\mu}^\pm)} \tilde{Q}_{\hat{\mu}^\pm}(\tfrac{1}{2} \mathbf{t}_B),
 \end{eqnarray}
 where, restricting the independent variables to $\mathbf{t}_B = [\mathbf{x}]_B$ as in (\ref{t_norm_power_sum}), we recover the usual Q Schur polynomials
 \begin{equation}
 Q(\mathbf{x}) = \tilde{Q}(\mathbf{t}_B),
 \end{equation}
 which, by (\ref{kappa_alpha_pfaff}), are the Pfaffians of the principal minors of the same skew symmetric matrix 
 ${\mathcal Q}_{\mu^\pm_j, \mu^\pm_k}({\bf t}_B)$. 
 
 Eq.~(\ref{bilinear_exp}) therefore reduces to the  result obtained in \cite{HO}, 
 expressing Schur functions as sums over products of Schur $Q$-functions,
 \begin{equation}
s_{(\alpha|\beta)}(\mathbf{t}')  = {(-1)^{\tfrac{1}{2}r(r+1)+s} \over 2^{2r-s}} \sum_{\mu\in {\mathcal P}(\alpha, \beta)} \sgn(\mu)
 (-1)^{\pi(\mu^-)+\tfrac{1}{2}m(\hat{\mu}^-)}
 \tilde{Q}_{\hat{\mu}^+}(\tfrac{1}{2}\mathbf{t}_B) \tilde{Q}_{\hat{\mu}^-}(\tfrac{1}{2}\mathbf{t}_B).
   \label{s_lambda_Q_bilinear_bis}
 \end{equation}
 \end{example}
 

 
 \begin{example}{\bf (Lattices of polynomial  KP vs. BKP $\tau$-functions)}.
\label{symm_polynom_bilin_sum}
A broad family of KP and BKP $\tau$-function lattices consisting of
more general (inhomogeneous) polynomials in the flow variables
is obtained if special restrictions are placed on the matrices $\tilde{A}$  relating them to those
appearing in (\ref{hat_g_exp_A}) and (\ref{hat_h_alpha_pm_exp_A}). 
Such  polynomial KP $\tau$-functions, their BKP analogs  and their fermionic representations were studied 
in \cite{Iv, HL, KvdL2, KvdLRoz, Roz}. They may be viewed as generalized Schur functions and Schur $Q$-functions,
which are expressible as finite linear combinations of ordinary Schur and Schur $Q$-functions.
We introduce the following fermionic operators
\begin{equation}
\psi_i(\tilde{A})=\hat{g}(\tilde{A})\psi_i \left( \hat{g}(\tilde{A)}\right)^{-1},\quad \psi_i^*(\tilde{A})=\hat{g}(\tilde{A})\psi_i^\dag \left( \hat{g}(\tilde{A})\right)^{-1}
\label{psi_A}
\end{equation}
where $\hat{g}(\tilde{A})$ is defined in (\ref{hat_g_exp_A}), and
\begin{equation}
\phi^\pm_i(A)=\hat{h}^\pm(A)\phi_i^\pm (\hat{h}^\pm(A))^{-1},
\label{phi_pm_A}
\end{equation}
where  $\hat{h}^\pm(A)$ is defined in (\ref{hat_h_alpha_pm_exp_A}).
\begin{lemma}
If the infinite skew matrix  $A$ is chosen to satisfy the antidiagonal triangular condition:
\begin{equation}
A_{j, -k} = 0 \quad \text{if } j>k
\end{equation}
and hence $\tilde{A}$ is has the upper  triangular form
\begin{equation}
\tilde{A}_{jk} = 
\begin{cases} (-1)^k A_{j, -k} \ \text{ if } \ j\le k \cr
0  \ \text{ if } \ j > k, 
\end{cases}
\end{equation}
then
\begin{equation}
\hat{g}(\tilde{A}) = \hat{h}^+(A) \hat{h}^-(A) 
\end{equation}
and
\begin{equation}
\psi_j(\tilde{A})=\tfrac{1}{\sqrt{2}}(\phi_j^+(A)- i \phi_j^-(A)),\quad \psi_{-j}^*(\tilde{A})=\tfrac{(-1)^j}{\sqrt{2}}(\phi_j^+(A)+ i \phi_j^-(A))
\end{equation}
\end{lemma}
\begin{proof}
This follows from the definitions (\ref{hat_g_exp_A}) and \ref{hat_h_alpha_pm_exp_A}) of   $\hat{g}(\tilde{A})$ and $\hat{h}^\pm(A)$,
the relation (\ref{neutral_charged}) between charged and neutral fermionic operators, the skew symmetry of the matrix $A$,
which leads to the cancellation of the sum over mixed terms $\phi^+_j\phi^-_k$ in the exponent,
and the mutual anticommutatitvity  of the two types of neutral fermi operators $\{\phi^+_j\}_{j\in \mathbf{Z}}$ and $\{\phi^-_j\}_{j\in \mathbf{Z}}$
\end{proof}

To define a lattice of polynomial KP $\tau$-functions that generalize the Schur polynomials we note that,
because  $\tilde{A}$ is upper triangular, $\hat{g}(h(A))$  stabilizes the vacuum $|0\rangle$
\begin{equation}
 \hat{g}(\tilde{A})|0\rangle =|0\rangle.
 \label{g_tilde_A_vac}
 \end{equation}
It follows  from (\ref{pi_alpha_beta_det}) that
\begin{equation}
\pi_{(\alpha|\beta)}(g(h(A)))({\bf t})  =
\mathrm {det} \left( \langle 0|\hat{\gamma}_+(\mathbf{t})\hat{g}(\tilde{A}) \psi_{\alpha_i}\psi^\dag_{-\beta_j-1}|0\rangle \right)_{i,j=1,\dots,r}
= \langle 0|\hat{\gamma}_+(\mathbf{t})\hat{g}(\tilde{A})|\lambda\rangle.
\label{pi_alpha_beta_def}
\end{equation}
Since both $\hat{\gamma}_+(\mathbf{t})$ and $\hat{g}(\tilde{A})$ stabilize the vacuum, we may define
\begin{eqnarray}
s_\lambda({\mathbf{t}}|\tilde{A})&\& := \pi_{(\alpha|\beta)}(g(h(A)))({\bf t}) \cr
&\&= (-1)^{\sum_{j=1}^r\beta_j}(-1)^{\tfrac{1}{2}r(r-1)} \langle 0|\hat{\gamma}_+(\mathbf{t})\psi_{\alpha_1}(\tilde{A})\cdots \psi_{\alpha_r}(\tilde{A}) \psi^*_{-\beta_1-1}(\tilde{A})\cdots \psi^*_{-\beta_r-1}(\tilde{A})|0\rangle
\cr
&\&= (-1)^{\sum_{j=1}^r\beta_j}\mathrm {det}\left( \langle 0|\hat{\gamma}_+(\mathbf{t})\psi_{\alpha_i}(\tilde{A}) \psi^*_{-\beta_j-1}(\tilde{A})|0\rangle \right)_{i,j=1,\dots, r}, 
\label{gen_Schur_def}
\end{eqnarray}
and these can be interpreted as generalized Schur functions (cf.~eqs.~(\ref{lambda_frob_vac}), \ref{schur_fermionic})).


Note that,  as for ordinary Schur functions \cite{Mac}, we have, for any ${\bf p}=(p_1, p_2, \dots )$, 
 \begin{equation}
 s_{\lambda}({\mathbf{t} +{\bf p}}|\tilde{A})=\sum_{\rho\subseteq \lambda} 
  s_{\lambda/\rho}({\bf p}|\tilde{A}) s_{\rho}(\mathbf{t}), 
  \label{s_lambda_A_p_shifted}
 \end{equation}
 where the generalized skew Schur function is defined fermionically as
\begin{equation}
s_{\lambda/\rho}({\mathbf{t}}|\tilde{A}):= (-1)^{\sum_{j=1}^r\beta_j}(-1)^{\tfrac{1}{2}r(r-1)}
\langle \rho|\hat{\gamma}_+(\mathbf{t})\psi_{\alpha_1}(\tilde{A})\cdots 
\psi_{\alpha_r}(\tilde{A})\psi^*_{-\beta_1-1}(\tilde{A}) \cdots \psi^*_{-\beta_1-1}(\tilde{A})|0\rangle.
\label{general_schur_translate}
\end{equation}
This follows from factorizing
 \begin{equation}
 \hat{\gamma}_+({\bf t} + {\bf p}) =  \hat{\gamma}_+ ({\bf t})  \hat{\gamma}_+({\bf p} )
 \end{equation}
 in (\ref{pi_alpha_beta_def}) and inserting the projection operator 
 \begin{equation}
 \sum_{\rho} | \rho;0 \rangle \langle \rho;0 |
 \end{equation}
onto the  $n=0$ fermionic charge sector between the factors.
It follows from (\ref{general_schur_translate}) that   
$s_{\lambda}({\mathbf{t}}|\tilde{A})$ is, indeed, a polynomial in the variables ${\bf t}=(t_1, t_2, \dots)$
since setting ${\bf p} ={\bf 0}$ in (\ref{s_lambda_A_p_shifted})  gives
\begin{equation}
 s_{\lambda}({\mathbf{t} }|\tilde{A})=\sum_{\rho\subseteq \lambda}  s_{\lambda/\rho}({\bf 0}|\tilde{A}) s_{\rho}(\mathbf{t}).
 \label{s_lambda_A_expan_s_rho}
 \end{equation}


Next,  we define the (scaled) generalized Schur $Q$-functions to be
\begin{eqnarray}
{\mathcal Q}_{\hat{\mu}^\pm}({\mathbf{t}_B}|A)&\&:= \kappa_{\alpha} (h(A))({\bf t}_B) =
\langle 0|\gamma^{B\pm}(\mathbf{t}_B)\phi^\pm_{\hat{\mu}^\pm_1}(A)\cdots \phi^\pm_{\hat{\mu}^\pm_{m(\hat{\mu}^\pm)}}(A) |0\rangle
 \cr
&\&=\mathrm {Pf} \left( \langle 0|\gamma^{B^\pm}(\mathbf{t}_B)\phi^\pm_{\hat{\mu}^\pm_i}(A^\pm)\phi^\pm_{\hat{\mu}^\pm_j}(A)|0\rangle \right)_{i,j=1,\dots, m(\hat{\mu}^\pm)} \cr
&\& =: 2^{-\tfrac{1}{2} m(\hat{\mu}^\pm)} \tilde{Q}_{\hat{\mu}^\pm}(\tfrac{1}{2}{\mathbf{t}_B}|A)
\label{gen_Q_def}
\end{eqnarray}
(Recall that $\hat{\mu}^\pm$ denotes the {\em  supplemented partition}, 
of even cardinality, which is obtained from  $\mu^\pm$ by adding a $0$ part if it has odd cardinality.)
These too are polynomials in the BKP flow variables $\mathbf{t}_B$ since, like (\ref{s_lambda_A_expan_s_rho}), we 
may express ${\mathcal Q}_{\hat{\mu}^\pm}({\mathbf{t}_B}|A)$ as a finite linear combination of the Schur $Q$-functions
${\mathcal Q}_{\hat{\nu}}({\mathbf{t}_B})$.
\begin{equation}
{\mathcal Q}_{\hat{\mu}^\pm}(\mathbf{t}_B | A)({\bf p}_B)) =
\sum_{\nu \subseteq \hat{\mu}^\pm}\, 
 \,{\mathcal Q}_{\hat{\mu}^\pm/\nu}({\bf 0}|A)\,{\mathcal Q}_{\nu}({\mathbf{t}_B}),
\end{equation}

 Restricting to this particular case of Theorem \ref{KP_BKP_bilinear} gives:
\begin{corollary} 
\label{bilinear_general_schur}
The generalized Schur function $s_{(\alpha|\beta)}$  (\ref{gen_Schur_def}), evaluated  at $\mathbf{t}=\mathbf{t}'$, 
may be  expressed as the following sum  over products $ \tilde{Q}_{\hat{\mu}^+} \tilde{Q}_{\hat{\mu}^-}$ of pairs of generalized 
$Q$ Schur functions (\ref{gen_Q_def}):
 \begin{equation}
s_{(\alpha|\beta)}(\mathbf{t}'|A)  = {(-1)^{\tfrac{1}{2}r(r+1)+s} \over 2^{2r-s}} \sum_{\mu\in {\mathcal P}(\alpha, \beta)} \sgn(\mu)
 (-1)^{\pi(\mu^-)+\tfrac{1}{2}m(\hat{\mu}^-)}
 \tilde{Q}_{\hat{\mu}^+}(\tfrac{1}{2}\mathbf{t}_B|A) \tilde{Q}_{\hat{\mu}^-}(\tfrac{1}{2}\mathbf{t}_B|A).
   \label{s_alpha_beta_Q_bilinear}
 \end{equation}
 \end{corollary}
 \begin{remark}
To interpret the results in terms of symmetric polynomials in a set of auxiliary variables,
 and consistently be able to choose the specialization $\mathbf{t} = \mathbf{t}'$ in (\ref{s_alpha_beta_Q_bilinear}),
we must reinterpret them as supersymmetric power sums,  as in (\ref{t_prime_y_sum}).
 \end{remark}
 
A particularly simple case  consists of the double $D(\alpha)$ of a strict partition $\alpha$, 
for which the Frobenius indices $(\alpha |\beta)$ are related by $\alpha= I(\beta)$.
Since the only polarization in this case is itself, the sum consists of just one term, and we obtain
(as for ordinary Schur functions)
 \begin{equation}
 s_{D(\alpha)}(\mathbf{t}'|A)= 2^{-r} \left( \tilde{Q}_{{\hat\alpha}}(\tfrac{1}{2}\mathbf{t}_B|A^\pm) \right)^2.
 \end{equation}
\end{example}


 
 \begin{example}{\bf (Lattices of polynomial $\tau$-functions parametrized by ${\bf r}$ and ${\bf p}$)}.
\label{r_p_polynom_bilin_sum}
An interesting  subclass of generalized Schur functions and Schur $Q$-functions
is obtained if we choose the skew matrix $A$ to be parametrized in terms    
two sets of  infinite parameters ${\bf r}:=\{r_j\}_{j \in \mathbf{Z}}$ and  ${\bf p}=(p_1,p_2,p_3,\dots)$,
where we assume the relations
\begin{equation}
r_j= r_{1-j}, \quad j\in \mathbf{N}.
\end{equation}

Let 
\begin{eqnarray}
\hat{A}^\mathbf{r}_j &\&:=\sum_{k\in\mathbf{Z}} \psi_k \psi^\dag_{k+j} r_{k+1}\cdots r_{k+j}, \quad j=1,2,3,\dots
\label{hat_A_r_j} \\
\hat{A}^{\mathbf{r}\pm}_j&\&:=\sum_{k\in\mathbf{Z}}(-1)^{k+1} \phi^\pm_k \phi^\pm_{-k-j} r_{k+1}\cdots r_{k+j},\quad j=1,3,5,\dots
\label{hat_A_j^pm} \\
\hat{A}^\mathbf{r}(\mathbf{p}) &\&:=\sum_{j>0}p_j A^\mathbf{r}_j =: \sum_{j, k  \in \mathbf{Z}} \tilde{A}^\mathbf{r}(\mathbf{p})_{jk} \no{\psi_j \psi^\dag_k}, \\
 \hat{A}^{\mathbf{r} \pm}(\mathbf{p}_B)&\&:=\sum_{j>0,\,{\rm odd}}p_j\hat{A}^{\mathbf{r} \pm}_j =: \tfrac{1}{2}\sum_{j,k \in \mathbf{Z}}  A^{\mathbf{r}}(\mathbf{p}_B)_{jk} \no{\phi^\pm_j \phi^\pm_{k}} 
\label{hat_A_r_p}
\end{eqnarray}
where
\begin{eqnarray}
 \tilde{A}^\mathbf{r}(\mathbf{p})_{jk}&\&= p_{k-j} r_{j+1} \cdots r_k, \quad k>j 
 \label{A_r_p_jk},
   \\
 &\&\cr
A^{\mathbf{r}}(\mathbf{p}_B)_{jk} &\&=\begin{cases}  (-1)^k \, p_{-j-k}\, r_{j+1} \cdots r_{-k} , \quad j+k < 0 
  \text{ and odd}\cr
 \qquad  0 \quad \text{ if } j+k \ \text{ is even or } j+k \ge 0.
  \end{cases}
  \cr
  &\&
 \label{A_r_p_B_jk}
\end{eqnarray}
and define the group elements
\begin{eqnarray}
\hat{g}(\tilde{A}^\mathbf{r}(\mathbf{p})) := e^{\hat{A}^\mathbf{r}({\bf p}) }, \quad \hat{h}^\pm(A^{\mathbf{r}}(\mathbf{p}_B)):= e^{\hat{A}^{\mathbf{r}\pm }(\mathbf{p}_B) }.
\label{g_Ar_h_Ar_def}
\end{eqnarray}
 We then have:
 \begin{proposition}
\begin{equation}
s_\lambda(\mathbf{t} | \tilde{A}^\mathbf{r}(\mathbf{p}))  = \pi_{(\alpha|\beta)}(g(\tilde{A}^\mathbf{r}(\mathbf{p})))({\bf t}) =
\,\sum_{\rho \subseteq \lambda}
\,r_{\lambda/\rho}\,s_{\lambda/\rho}({\bf p})\,s_{\rho}({\bf t}),
\label{s_lambda_r_p_expansion}
\end{equation}
where $s_{\lambda/\rho}({\bf p})$ is the skew Schur function corresponding to the skew
partition $\lambda/\rho$, and
\begin{equation}
 r_{\lambda/\rho} :=\prod_{i=1}^{\ell(\lambda)} \prod_{j=\rho_i+1}^{\lambda_i}r_{j-i} ={ r_\lambda \over r_\rho}
 \label{skew_content_prof_r}
\end{equation} 
is the content product over nodes of the skew  Young diagram of $\lambda/\rho$, and
\begin{equation}
{\mathcal Q}_{\hat{\mu}^\pm}(\mathbf{t}_B | A^{\mathbf{r}}({\bf p}_B)) = \kappa_{\hat{\mu}^\pm}(g(A^{\mathbf{r}} (\mathbf{p}_B)))({\bf t}_B) =
\sum_{\nu \subseteq \hat{\mu}^\pm}\, 
r^B_{\mu^\pm /\nu} \,{\mathcal Q}_{\hat{\mu}^\pm/\nu}({\bf p}_B)\,{\mathcal Q}_{\nu}({\mathbf{t}_B}),
\label{Q_mu_r_p_expansion}
\end{equation}
 where 
\begin{equation}
{\mathcal Q}_{\hat{\mu}^\pm/\nu}({\bf p}_B) = (\nu^\pm|\gamma^{B\pm}(\mathbf{p}_B) | \hat{\mu}^\pm)
\end{equation}
  is the (scaled) skew Schur Q-function 
 corresponding to the skew partition $\mu^\pm/\nu$ 
  and
\begin{equation}
 r^B_{\mu^\pm/\nu}   :=\prod_{i=1}^{m(\mu^\pm)} \prod_{j=\nu_i+1}^{\mu^\pm_i} r_j ={ r^B_{\mu^\pm} \over r^B_\nu}
 \label{skew_strict_content_prod_r}
\end{equation} 
is the content product over nodes of the {\em shifted} skew Young diagram  \cite{Iv} of $\mu^\pm/\nu$.

 \end{proposition}
\begin{proof}This follows along the lines of  \cite{Or1, OS}.
Define the operators
\begin{equation}
\hat{C}^\mathbf{r}:=\exp \left( \sum_{j< 0} T^\mathbf{r}_{j}\psi^\dag_j\psi_j -\sum_{i\ge 0} T^\mathbf{r} _j\psi_j\psi^\dag_j \right) 
\label{C_r}
\end{equation}
where
\begin{equation}
r_j=e^{T^\mathbf{r}_{j-1}-T^\mathbf{r}_{j}}, \quad T^\mathbf{r}_0:=0.
\label{r_T}
\end{equation}
Imposing the restriction
\begin{equation}
T^\mathbf{r}_j=-T^\mathbf{r}_{-j},
\label{B-reduction}
\end{equation}
 implies
\begin{equation}
r_j=r_{1-j}.
\label{B-reduction-r}
\end{equation}
We also define 
\begin{equation}
\hat{C}^{\mathbf{r}\pm} := \exp \sum_{j\ge 0} (-1)^{j+1}T^\mathbf{r}_j\phi^\pm_j\phi^\pm_{-j} .
\label{C_r_pm}
\end{equation}
It then follows from  (\ref{neutral_charged}) that
\begin{equation}
\label{C_C_pm}
\hat{C}^\mathbf{r}=\hat{C}^{\mathbf{r}+}\hat{C}^{\mathbf{r}-}=\hat{C}^{\mathbf{r}-}\hat{C}^{\mathbf{r}+}.
\end{equation}

From (\ref{psi_i_anticomm}) and (\ref{neutral_ccr}) we get
\begin{equation}
\hat{C}^\mathbf{r}\psi_j (\hat{C}^\mathbf{r})^{-1}= e^{-T^\mathbf{r}_j}\psi_j,\quad \hat{C}^\mathbf{r} \psi^\dag_j (\hat{C}^\mathbf{r})^{-1}= e^{T^\mathbf{r}_j}\psi^\dag_j, \quad 
\hat{C}^{\mathbf{r}\pm} \phi^\pm_j (\hat{C}^{\mathbf{r}\pm})^{-1}= e^{-T^\mathbf{r}_j}\phi^\pm_j
\end{equation}
It follows that the operators defined in (\ref{hat_A_r_j}, (\ref{hat_A_j^pm}), (\ref{hat_A_r_p}) are given by
\begin{equation}
\hat{A}^\mathbf{r}_j: = \hat{C}^\mathbf{r} J_j(\hat{C}^\mathbf{r})^{-1},\quad j=1,2,\dots
\end{equation}
and
 \begin{equation}
\hat{A}^{\mathbf{r}\pm}_j :=\hat{C}^{\mathbf{r}\pm} J_j^{B\pm} (\hat{C}^{\mathbf{r}\pm})^{-1},\quad j=1, 3, 5,\dots
\end{equation}
For odd $j$  we have
\begin{equation}
\hat{A}^\mathbf{r}_j  =  \hat{C}^\mathbf{r}(J_j^{B+}+J^{B-}_j) (\hat{C}^\mathbf{r})^{-1}=\hat{A}_j^{\mathbf{r}+} + \hat{A}_j^{\mathbf{r}-},\quad i=1,3,5,\dots
\end{equation}

Defining
\begin{equation}
\hat{\gamma}^{\mathbf{r}}_+({\bf p}):=(\hat{C}^\mathbf{r})^{-1}\hat{\gamma}_+({\bf p})\hat{C}^\mathbf{r},\quad 
\hat{\gamma}^{\mathbf{r},B\pm}({\bf p}_B):= (\hat{C}^{\mathbf{r}\pm})^{-1}\hat{\gamma}^{B\pm}({\bf p}_B) \hat{C}^{\mathbf{r} \pm}
\end{equation}
it follows from  (\ref{C_C_pm}) and (\ref{gamma_factoriz}) that
\begin{equation}
\label{gamma_r_factoriz}
\hat{\gamma}_+^\mathbf{r}({\bf p}')=\hat{\gamma}^{\mathbf{r} ,B+}({\bf p}_B)\gamma^{ \mathbf{r}, B-}({\bf p}_B).
\end{equation}
The equalities
\begin{equation}
s_\lambda({\bf t}|A^\mathbf{r}({\bf p}))=\langle 0|\hat{\gamma}_+(\mathbf{t})\hat{\gamma}^\mathbf{r}_+({\bf p})|\lambda\rangle
\label{gen-S=VEV}
\end{equation}
and
\begin{equation}
{\mathcal Q}_{\mu^\pm}({\bf t}_B|A^{\mathbf{r}}({\bf p}_B))=\langle 0|\hat{\gamma}^\pm(\mathbf{t}_B)\hat{\gamma}^{\mathbf{r}, \mathbf{B}\pm}({\bf p}_B)|\mu^\pm\rangle.
\label{Q_r_p_fermi}
\end{equation}
follow from  definitions (\ref{gen_Schur_def}) and (\ref{gen_Q_def}), when
the group elements  $\hat{g}(h(A))$ and $\hat{h}^\pm(A)$  are chosen as in (\ref{g_Ar_h_Ar_def}) and $\psi_i(\tilde{A})$
and $\phi^\pm_j(A)$ computed by inserting these in (\ref{psi_A}) and (\ref{phi_pm_A}).

Inserting the projection operator 
\begin{equation}
\Pi_0=\sum_{\rho\in\mathop\mathrm{P}\nolimits} |\rho; 0\rangle \langle\rho; 0|
\end{equation}
onto the subspace ${\mathcal F}_0\subset{\mathcal F}$  between $\hat{\gamma}_+({\bf t})$ 
and $\hat{\gamma}^\mathbf{r}_+({\bf p})$  in (\ref{gen-S=VEV}), and using 
\begin{equation}
s_\lambda(\mathbf{t})=\langle 0|\hat{\gamma}_+(\mathbf{t})|\lambda\rangle
\end{equation}
and the slightly more general relation
\begin{equation}
r_{\lambda/\rho}s_{\lambda/\rho}({\bf p})= \langle \rho|\hat{\gamma}_+^{\mathbf{r}}({\bf p}) |\lambda \rangle 
\end{equation}
gives (\ref{s_lambda_r_p_expansion}). (See formula (3.5.7) in  \cite{Or2}) 

Similarly, to prove (\ref{Q_mu_r_p_expansion}),  insert the projection operators 
\begin{equation}
\Pi^\pm = \sum_{\nu^{\pm}}  |\nu^{\pm} ) ( \nu^{\pm} |   
\end{equation}  
onto the subspaces $\tilde{{\mathcal F}}^\pm_\phi \subset {\mathcal F}$  between
$\hat{\gamma}^{B\pm}(\mathbf{t}_B)$ and $\gamma^{\mathbf{r}, B\pm}(\mathbf{p}_B)$  in (\ref{Q_r_p_fermi}) 
and use
\begin{equation}
{\mathcal Q}_{\mu^{\pm}}({\bf t}_B)=
\langle 0| \hat{\gamma}^{B\pm}({\bf t}_B)|\mu^{\pm} )
\label{Q-You}
\end{equation}
and
\begin{equation}
\label{skewQ-fermionic}
r^B_{\mu^\pm /\nu}{\mathcal Q}_{\mu^\pm/\nu}({\bf p}_B)=(\nu^\pm| \hat{\gamma}^{\mathbf{r}, B\pm}({\bf p}_B)|\mu^\pm).
\end{equation}
\end{proof}
It then follows  from (\ref{gamma_r_factoriz}) that  the polynomials 
$s_\lambda({\bf t}'|A^\mathbf{r}({\bf p}'))$ and $ \tilde{Q}_{\hat{\mu}^\pm}(\frac{1}{2}{\bf t}_B|A^{\mathbf{r} \pm}({\bf p}_B))$ are related
by (\ref{s_alpha_beta_Q_bilinear}), with $\tilde{A}= \tilde{A}^\mathbf{r}(\mathbf{p}')$ and $A= A^\mathbf{r}(\mathbf{p}_B)$.
\begin{remark}
With the special choice 
\begin{eqnarray}
\mathbf{p}' &\& =\mathbf{t}_0:=(1,0,0,\dots)
\label{p_0_def}
\\
t_j &\&={1\over j}\sum_{k=1}^N x_k^j
\label{t-x}
\cr
r_j&\&=r_j(z,z'):=-(z+j)(z'+j),
\label{r_j_laguerre}
\end{eqnarray}
eq.~(\ref{s_lambda_r_p_expansion}) gives the multivariate Laguerre polynomial  \cite{Ol, La}. 
To see this, substitute (\ref{p_0_def}),  (\ref{r_j_laguerre}) into eq.~(\ref{s_lambda_r_p_expansion}) to obtain 
\begin{equation}
s_\lambda(\mathbf{t}|A^{\mathbf{r}(z,z')}(\mathbf{t}_0))= 
\sum_{\rho\subseteq\lambda} (-1)^{|\lambda|-|\rho|}(z)_{\lambda/\rho}(z')_{\lambda/\rho}
s_{\lambda/\rho}(\mathbf{t}_0)s_{\rho}(\mathbf{t}),
\label{multivar_laguerre_exp}
\end{equation}
where
\begin{equation}
(z)_{\lambda/\rho}:=\prod_{(i,j)\in\lambda/\rho }(z+j-i)
\end{equation}
is the content product of $\{z+j\}_{j\in \mathbf{Z}}$ over the skew Young diagram  $\lambda/\rho$, and 
\begin{equation}
s_{\lambda/\rho}(\mathbf{t}_0)= \mathrm {det}\left({1\over (\lambda_i - i -\rho_j +j)!}\right)_{1\le i, j \le \ell(\lambda)}.
\end{equation}
Eq.~(\ref{multivar_laguerre_exp}) coincides with formula (4.3) of \cite{Ol} for multivariate Laguerre polynomials.
where the coefficients were also written in terms of integer evaluations of shifted Schur functions \cite{OkOl}.
\end{remark}

Note that  an arbitrary choice of parameters  $(z,z')$  in  formula (\ref{multivar_laguerre_exp}) does not satisfy
 the relation (\ref{g_h_def}), which is needed in order to apply  Corollary \ref{bilinear_general_schur}. 
For this, the reduction condition (\ref{B-reduction-r}) must be satisfied,
which requires that we impose the constraint \hbox{$z'=-1-z$}, giving
\begin{equation}
r_j=(z+j)(z+1-j) = r_{1-j}.
\end{equation}
Substituting this into (\ref{skew_strict_content_prod_r}), we get eq.~(\ref{Q_mu_r_p_expansion}), with
\begin{equation}
r^B_{\mu^\pm/\nu}(z) :=\prod_{i=1}^{m(\nu)}\prod_{j=\nu_i+1}^{\mu_i}(z+j)(1+z-j)
\end{equation}
and 
\begin{equation}
\tilde{Q}_{\mu/\nu}(\tfrac 12\mathbf{t}_0)
= \mathrm {Pf} \left(\frac{\mu_i-\mu_j-\nu_i+\nu_j}{(\mu_i+\mu_j-\nu_i-\nu_j)(\mu_i-\nu_i)!(\mu_j-\nu_j)!}\right)_{1\le i,j \le m(\mu)},
\end{equation}
with the understanding that $\nu_i=0$ if $i > m(\nu)$. 
 \end{example}
 

\medskip
\noindent 
\small{{\em Acknowledgements.} The work of J.~H. was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC); that of A.~Yu.~O.  by RFBR grant 18-01-00273a and state assignment  N 0149-2019-0002. \hfill

 
\begin{center}
\smallskip
This work is dedicated to {\bf Grigori Olshanski} on the occasion of his 70th birthday. \hfill
\end{center}



  







 
