Bilinear expansions of KP multipair correlators in BKP correlators

A. Yu<EMAIL_ADDRESS>

Institute of Oceanology, Nahimovskii Prospekt 36, Moscow 117997, Russia, and
National Research University Higher School of Economics, Moscow, Russia

Dedicated to John Harnad on the occasion of his 75th birthday

###### Abstract

In earlier work, Schur lattices of KP and BKP $\tau$-functions, denoted
$\pi_{\lambda}(g)({\bf t})$ and $\kappa_{\alpha}(h)({\bf t}_{B})$,
respectively, defined as fermionic vacuum expectation values, were associated
to every $\operatorname{GL}(\infty)$ group element $\hat{g}$ and
$\operatorname{SO}(\tilde{{\mathcal{H}}}^{\pm},Q_{\pm})$ group element
$\hat{h}$. The elements of these lattices are labelled by partitions $\lambda$
and strict partitions $\alpha$, respectively. It was shown how the former may
be expressed as finite bilinear sums over products of the latter. In this
work, we show that two-sided KP tau functions corresponding to any given
$\hat{g}$ may similarly be expressed as bilinear combinations of the
corresponding two-sided BKP tau functions.

##  1 Introduction

This work is a continuation of [9],[10],[11],[30] which concerned bilinear
expansions of Schur lattices $\\{\pi_{\lambda}(g)({\bf t})\\}$ of KP
$\tau$-functions, labelled by partitions $\lambda$ and
$\operatorname{GL}(\infty)$ group elements $\hat{g}$ expressible as sums of
products of corresponding lattices $\\{\kappa_{\alpha}(h)({\bf t}_{B})\\}$ of
BKP $\tau$ functions, labelled by strict partitions $\alpha$ and
$\operatorname{SO}(\infty)$ group elments $\hat{h}$.

The approach was based on the notion of tau function, as introduced by Sato
[33] expressed as vacuum state expectation values (VEV ’s) of products free
fermionic operators, and their exponentials, as developed in the works of
Kyoto school, [2],[3],[6] .

###  1.1 Few words about the problem

Definitions will be given later below in Section 2.

#### (a) Schur functions and the projective Schur functions.

The following relation is known (for instance, see [17]):

$\left(Q_{\alpha}({\mathbf{p}^{\rm B}})\right)^{2}=s_{D(\alpha)}(\mathbf{p}),$
(1.1)

where $Q_{\alpha}$ is the projective Schur function written as the function of
power sum variables

${\mathbf{p}^{\rm B}}=(p^{\rm B}_{1},p^{\rm B}_{3},p^{\rm B}_{5},\dots)$ (1.2)

and where $s_{\lambda}$ is the Schur function written as function of power sum
variables

$\mathbf{p}=(p_{1},p_{2},p_{3},p_{4},p_{5},\dots)$ (1.3)

and in formula (1.1) these two sets of variables are related as follows:

$\mathbf{p}=\mathbf{p}^{\prime}:=(p_{1},0,p_{3},0,p_{5},0,\dots)$ (1.4)

where

$p_{i}=2p^{\rm B}_{i},\quad i\,\,{\rm odd}$ (1.5)

where $2p^{\rm B}_{n}=p_{n}$ and are $p^{\rm B}_{n}$ and $p_{n}$ are commonly
related to the two different sets of the variables, say, respectively,
$z_{1},\dots,z_{N}$ and $x_{1},\dots,x_{M}$ as Newton sums:

$p^{\rm B}_{2m-1}=p_{2m-1}(z_{1},z_{2},\dots)=\sum_{i=1}^{N}z_{i}^{2m-1},\quad
m=1,2,3,\dots$ (1.6)
$p_{m}=p_{m}(x_{1},\dots,x_{M})=\sum_{i=1}^{M}x_{i}^{m},\quad m=1,2,3,\dots$
(1.7)

and therefore are called power sum variables. The projective Schur function is
a polynomial quasihomogeneous function in the power sum variables and
symmetric homogenious polynomial in the variables
$\mathbf{z}=(z_{1},\dots,z_{N(\mathbf{z})})$, related to the power sums by
(1.6). It is labeled by a strict partition (multiindex)
$\alpha=(\alpha_{1},\dots,\alpha_{k})$ (where
$\alpha_{1}>\alpha_{2}>\cdots>\alpha_{k}\geq 0$ is the set of integers). We
will recall the definition later in the text. We will write the projective
Schur function either as symmetric function $Q_{\alpha}(\mathbf{z})$ or as
polynomial $Q_{\alpha}({\mathbf{p}^{\rm B}})$ and we hope it does not produce
a misunderstanding.

$s_{D(\alpha)}(\mathbf{p})$ in the right hand side is the Schur function
labeled by a special partition (the multiindex) denoted $D(\alpha)$ which is
called the double of the strict partition $\alpha$. In general, the Schur
function $s_{\lambda}(\mathbf{p})$ is defined for any partition $\lambda$ and
is a quasihomogenious polynomial in the variables
$(p_{1},p_{2},p_{3},\dots)=:\mathbf{p}$ where in contrast to (1.4) the
variables indexed with even numbers are also in presence. However the Schur
function in the right hand side of (1.1) is evaluated for the restricted set
of variables denoted $\mathbf{p}^{\prime}$ which is
$(p_{1},0,p_{3},0,p_{5},0,\dots)$ and $\lambda$ is chosen to be the double of
$\alpha$. In what follows we also use the so-called Frobenius coordinated of
$\lambda$, $\lambda=(\alpha|\beta)$ where
$\alpha=(\alpha_{1},\dots,\alpha_{r})$ and $\beta=(\beta_{1},\dots,\beta_{r})$
is the pair of strict partitions. The function $s_{\lambda}$ is a symmetric
homogenious polynomial in the variables $\mathbf{x}$ and commonly is written
as $s_{\lambda}(\mathbf{x})$ which is $s_{\lambda}(\mathbf{p}(\mathbf{x}))$
(we hope it will not produce the incomprehension).

Equality (1.1) is the fundamental equality which relates Schur functions and
projective Schur functions, to our knowledge at first it was found in [37].

Consider two independent sets of variables $\mathbf{a}=(a_{1},\dots,a_{N})$
and $\mathbf{b}=(b_{1},\dots,b_{N})$. (Sometimes we will use the notation
$N(\mathbf{a})$ for the number of the variables in the set $\mathbf{a}$.)

Let us choose

$p_{n}=p_{n}(\mathbf{a}/-\mathbf{b}):=\sum_{i=1}^{N}\left(a_{i}^{n}-(-b_{i})^{n}\right),\quad
n=1,2,3,\dots\,.$ (1.8)

Such $\mathbf{p}(\mathbf{a}/-\mathbf{b})$ we will call supersymmetric Newton
sums of the variables $\mathbf{a},\mathbf{b}$. Below, we use the notation
$\mathbf{p}(\mathbf{a}/-\mathbf{b})=\left(p_{1}(\mathbf{a}/-\mathbf{b}),p_{2}(\mathbf{a}/-\mathbf{b}),p_{3}(\mathbf{a}/-\mathbf{b}),\dots\right)$
and
$s_{\lambda}(\mathbf{a}/-\mathbf{b}):=s_{\lambda}(\mathbf{p}(\mathbf{a}/-\mathbf{b}))$
(the similar notation is used in Ex 23 Chapter I, 3 of [17]).

Notice that $s_{\lambda}(\mathbf{a}/-\mathbf{a})$ can be written as
$s_{\lambda}(\mathbf{p}^{\prime})$ where
$p^{\prime}_{2n-1}=2\sum_{i=1}^{N}a_{i}^{2n-1}$. We will see that the
“sypersymmetric Newton sums” are quite natural for our problems and we shall
use it throughout the paper.

######  Remark 1.1.

One an easily conjecture that for any set $\mathbf{p}=(p_{1},p_{2},\dots)$
there exits such a number $N$ (perhaps, infinite) and two sets
$\mathbf{a}=(a_{1},\dots,a_{N})$ and $\mathbf{b}=(b_{1},\dots,b_{N})$ that
(1.8) is true.

In the present work we find the generalization of (1.1) which symbolically may
be written as:

$s_{(\alpha|\beta)}(\mathbf{a}/-\mathbf{b})=\sum_{m(\zeta^{-}),N(\mathbf{z}^{-})\atop\zeta^{-},\zeta^{+},\mathbf{z}^{-},\mathbf{z}^{+}}\left[\zeta^{+},\zeta^{-}\atop\alpha,\beta\right]\left[\mathbf{z}^{+},\mathbf{z}^{-}\atop\mathbf{a},\mathbf{b}\right]Q_{\zeta^{+}}\left(\mathbf{z}^{+}\right)Q_{\zeta^{-}}\left(\mathbf{z}^{-}\right)$
(1.9)

where the partition $\lambda=(\alpha|\beta)$ and the sets of complex numbers
$\mathbf{a}$,$\mathbf{b}$ are free. The sum in the right hand side is taken
over the natural numbers $m(\zeta^{-}),N(\mathbf{z}^{-})$, over the strict
partitions $\zeta^{\pm}$ and over the sets of (complex) numbers
$\mathbf{z}^{\pm}$:

$\zeta^{\pm}=(\zeta^{\pm}_{1},\dots,\zeta^{\pm}_{m(\zeta^{\pm})}),\quad\mathbf{z}^{\pm}=(z^{\pm}_{1},\dots,z^{\pm}_{N(\mathbf{z}^{\pm})}),$
(1.10)

where $m(\zeta^{+})+m(\zeta^{-})=2r,\,N(\mathbf{z}^{+})+N(\mathbf{z}^{-})=2N$,
and implies that $\zeta^{+}\cup\zeta^{-}=\alpha\cup(\beta+1)$ and
$\mathbf{z}^{+}\cup\mathbf{z}^{-}=\mathbf{a}\cup\mathbf{b}$ 222 it would be
symmetric for semiinteger partitions and the labels of the Fourier modes, but
we avoid semiintegers.  and the weights denoted by square brackets will by
written down in Section 4. The projective Schur functions in the right hand
side are written as symmetric functions in the variables
$\mathbf{z}^{\pm}=(z^{\pm}_{1},\dots,z^{\pm}_{N(\mathbf{z}^{\pm})})$, and
these variables are selected from the set
$a_{1},\dots,a_{N},b_{1},\dots,b_{N}$. The parts of the partitions
$\zeta^{\pm}=(\zeta^{\pm}_{1},\dots,\zeta^{\pm}_{m(\zeta^{\pm})})$ are
selected from the set
$\alpha_{1},\dots,\alpha_{r},\beta_{1}+1,\dots,\beta_{r}+1$.333 it would be
symmetric for semiinteger partitions where
$\alpha_{i}\to\alpha_{i}+\frac{1}{2},\,\beta_{i}\to\beta_{i}-\frac{1}{2}$ and
the labels of the Fourier modes $\psi_{i}\to\psi_{i+\frac{1}{2}}$, as it was
done in Kac’s papers, but we avoid semiintegers.

Actually, the case $\mathbf{a}=\mathbf{b}$ was already studied in our previous
work [9]. Let us note that under the pair of restrictions: $\alpha=\beta+1$
and $\mathbf{a}=\mathbf{b}$ we get formula (1.1) as the particular case of
(1.9).

#### (b) KP and BKP lattice tau functions.

Apart of the relation (1.9) we get much more general relation, that is a
relation between KP and BKP tau functions. Let us remined that the Schur
function is the simplest nontrivial example of the KP tau function [33] while
the projective Schur function is the simplest nontrivial example of the BKP
tau function [37],[25]. 444 Let us note that by BKP tau function we mean the
tau function introduced by Kyoto school in [2]. Another tau function also
called the BKP one was introduced in [14]. Both tau functions have a lot of
applications in various problems of mathematical physics, for instance in
random matrix theory.

The lattice KP tau function can be written as a sum over partitions as
follows:

$S_{\lambda}(\mathbf{p}|\hat{g})=\sum_{\mu\in\mathop{\mathrm{missing}}{P}\nolimits}s_{\mu}(\mathbf{p})\hat{g}_{\mu,\lambda}$
(1.11)

and depends on $\mathbf{p}=(p_{1},p_{2},\dots)$ and also on a partition (a
multiindex) $\lambda\in\mathop{\mathrm{missing}}{P}\nolimits$ where
$\mathop{\mathrm{missing}}{P}\nolimits$ is the set of all partitions.

Below we imply that $\mathbf{p}=\mathbf{p}(\mathbf{a}/-\mathbf{b})$. The
lattice BKP tau function can be written as

$K_{\mu}(\mathbf{z}|\hat{h})=\sum_{\nu\in\mathop{\mathrm{missing}}{DP}\nolimits}Q_{\nu}(\mathbf{z})\hat{h}_{\nu,\mu}\,.$
(1.12)

In these formulas $\hat{g}_{\mu,\lambda}$ and $\hat{h}_{\nu,\mu}$ are certain
coefficients given by the choice of KP and BKP tau functions, see Section 2.6
below. This choice may be treated as a choice of a certain element of the
Clifford group $\hat{g}$ in the KP case which is defined by a choice of
$\hat{g}\in\hat{GL}_{\infty}$ in the KP case (see (2.118) below) and by
$\hat{h}\in\hat{B}_{\infty}$ in the BKP case (formula (2.123) below)
(therefore we label the left hand sides of (1.11) and of (1.12) with these
symbols).

######  Remark 1.2.

If $\hat{g}=1$ then
$S_{\lambda}(\mathbf{p}|\hat{g}=1)=s_{\lambda}(\mathbf{p})$.

If $\hat{h}=1$ then $K_{\mu}(\mathbf{z}|\hat{h}=1)=Q_{\mu}(\mathbf{z})$.

######  Remark 1.3.

If $\lambda=0$ then $S_{\lambda=0}(\mathbf{p}|\hat{g})$ is usual (“one-side”)
KP tau function.

If $\mu=0$ then $K_{\mu=0}({\mathbf{p}^{\rm B}}|\hat{h})$ is usual (“one-
side”) BKP tau function.

In the present work it is supposed that
$\hat{g},\hat{h}^{\pm}\in\hat{B}_{\infty}$ and
$\hat{g}=\hat{h}^{-}\hat{h}^{+}$ and $\hat{g}=\hat{h}^{-}\hat{h}^{+}$.

Under parametrization (1.8) and the mentioned condition
$\hat{g}=\hat{h}^{-}\hat{h}^{+}\in\hat{B}_{\infty}$ explained in Section 2.6,
the relation between (1.11) and (1.12) is identical to the relation (1.9)
where $s_{\lambda}(\mathbf{a}/-\mathbf{b})$ is replaced by
$S_{\lambda}(\mathbf{p}(\mathbf{a}/-\mathbf{b})|\hat{g})$ of (1.11) and where
$Q_{\zeta^{\pm}}(\mathbf{z}^{\pm})$ are replaces by
$K_{\mu}(\mathbf{z}|\hat{h}^{\pm})$ of (1.12):

$S_{(\alpha|\beta)}(\mathbf{a}/-\mathbf{b}|\hat{h}^{+}\hat{h}^{-})=\sum_{m(\zeta^{-}),N(\mathbf{z}^{-})\atop\zeta^{+},\zeta^{-},\mathbf{z}^{+},\mathbf{z}^{-}}\left[\zeta^{+},\zeta^{-}\atop\alpha,\beta\right]\left[\mathbf{z}^{+},\mathbf{z}^{-}\atop\mathbf{a},\mathbf{b}\right]K_{\zeta^{+}}\left(\mathbf{z}^{+}|\hat{h}^{+}\right)K_{\zeta^{-}}\left(\mathbf{z}^{-}|\hat{h}^{-}\right)$
(1.13)

where the summation range and the weights denoted by square brackets are the
same as in (1.9). This is a subject of Theorem 4.3 in Section 2.6.

######  Remark 1.4.

In the spirit of the terminology common in soliton theory one may say that
relation (1.13) is the dressed by element $\hat{g}$ version of (1.9).

######  Remark 1.5.

In our previous work [10] we use different notations:

$\pi_{(\alpha|\beta)}(\hat{g})({\mathbf{p}})=S_{(\alpha|\beta)}(\mathbf{p}|\hat{g})$
(1.14) $\kappa_{\alpha}(\hat{h})(2{\mathbf{p}^{\rm
B}})=K_{\alpha}(2{\mathbf{p}^{\rm B}}|\hat{h})$ (1.15)

see Remarks 2.14 and 2.16 in Section 4.1.

######  Remark 1.6.

We call a KP lattice tau function polynomial in case for any $\lambda$, in
(1.11), there is only a finite number of terms in the right hand side.
Examples of the polynomial KP tau functions were presented in [8], [11].
Similarly we call a BKP lattice tau function polynomial if there is a finite
number of terms in the right hand side of (1.12) for any $\mu$. Polynomial BKP
tau functions were studied in [16].

As we mentioned (see Remark), the simplest example of the KP polynomial tau
function is the Schur function $s_{\lambda}(\mathbf{p})$. Other examples, like
characters of linear groups or Laguerre polynomials can be found in [8],[11]
respectively. The simplest example of the BKP polynomial tau function is the
projective $Q_{\mu}$-function. Other examples may be found in [11] and in the
reference list there.

#### (c) The relation between KP and BKP two-sided tau functions.

A two-sided KP tau function can be written as a double sum over partitions

$\tau(\mathbf{p},\tilde{\mathbf{p}}|\hat{g})=\sum_{\lambda,\mu}s_{\mu}(\mathbf{p})\hat{g}_{\mu,\lambda}s_{\lambda}(\tilde{\mathbf{p}}).$
(1.16)

Each two-sided KP tau function depends on two sets of higher times
$\mathbf{p}=(p_{1},p_{2},\dots)$ and
$\tilde{\mathbf{p}}=(\tilde{p}_{1},\tilde{p}_{2},\dots)$. In the present paper
we consider only the case $\mathbf{p}=\mathbf{p}(\mathbf{a}/-\mathbf{b})$
according to (1.8). Similarly, we choose $\tilde{\mathbf{p}}$ to be

$\tilde{p}_{n}=\tilde{p}_{n}(\tilde{\mathbf{a}}/-\tilde{\mathbf{b}})=\sum_{i=1}^{\tilde{N}}\left(\tilde{a}_{i}^{n}-(-\tilde{b}_{i})^{n}\right).$
(1.17)

We will write two-sided tau KP function as
$\tau(\mathbf{a}/-\mathbf{b},\tilde{\mathbf{a}}/-\tilde{\mathbf{b}}|\hat{g})$
for the choice (1.8) and (1.17).

The solution of these equations is defined by the choice of $\hat{g}$ which
gives rise to the coefficients $\hat{g}_{\mu,\lambda}$, in a way described in
[35]. The alternatice description uses the notation of Sato Grassmannian and
to speak about a point of the Grassmannian instead of $\hat{g}$. We will try
to avoid this notion in order not to convert a given clear problem of writing
down an explicit equlity to a part of geometry.

The two-sided BKP tau function can be written as a double sum over strict
partitions (definitions see below):

$\tau^{\rm B}({\mathbf{p}^{\rm B}},\tilde{\mathbf{p}}^{\rm
B}|\hat{h})=\sum_{\lambda,\mu}Q_{\mu}({\mathbf{p}^{\rm
B}})\hat{h}_{\mu,\lambda}Q_{\lambda}(\tilde{\mathbf{p}}^{\rm B}).$ (1.18)

It depends on two sets of higher times labeled with odd numbers:
${\mathbf{p}^{\rm B}}=(p^{\rm B}_{1},p^{\rm B}_{3},\dots)$ and
$\tilde{\mathbf{p}}^{\rm B}=(\tilde{p}^{\rm B}_{1},\tilde{p}^{\rm
B}_{3},\dots)$. In case, for sets $\mathbf{z}=(z_{1},\dots,z_{N(\mathbf{z})})$
and $\tilde{\mathbf{z}}=(z_{1},\dots,z_{N(\tilde{\mathbf{z}})})$ we have

$p^{\rm B}_{n}=p^{\rm
B}_{n}(\mathbf{z})=\sum_{i=1}^{N(\mathbf{z})}z^{n}_{i},\quad\tilde{p}^{\rm
B}_{n}=\tilde{p}^{\rm
B}_{n}(\tilde{\mathbf{z}})=\sum_{i=1}^{N(\tilde{\mathbf{z}})}\tilde{z}^{n}_{i}\,.$
(1.19)

We will write $\tau^{\rm B}({\mathbf{p}^{\rm B}}(\mathbf{z}),{\mathbf{p}^{\rm
B}}(\tilde{\mathbf{z}})|\hat{h})=:\tau^{\rm
B}(\mathbf{z},\tilde{\mathbf{z}}|\hat{h})$.

If we compare (1.11) with (1.16) (also (1.12) with (1.18)) we see that it is a
sort of Fourier transform.

The relation between two-sided KP and two-sided BKP tau functions is as
follows:

$\tau(\mathbf{a}/-\mathbf{b},\tilde{\mathbf{a}}/-\tilde{\mathbf{b}}|\hat{h}^{+}\hat{h}^{-})=\sum_{N(\mathbf{z}^{-}),N(\tilde{\mathbf{z}}^{-})\atop\mathbf{z}^{+},\mathbf{z}^{-},\tilde{\mathbf{z}}^{+},\tilde{\mathbf{z}}^{-}}\left[\mathbf{z}^{+},\mathbf{z}^{-}\atop\mathbf{a},\mathbf{b}\right]\left[\tilde{\mathbf{z}}^{+},\tilde{\mathbf{z}}^{-}\atop\tilde{\mathbf{a}},\tilde{\mathbf{b}}\right]\tau^{\rm
B}\left(\mathbf{z}^{+},\tilde{\mathbf{z}}^{+}|\hat{h}^{+}\right)\tau^{\rm
B}\left(\mathbf{z}^{-},\tilde{\mathbf{z}}^{-}|\hat{h}^{-}\right)$ (1.20)

The sum in the right hand side is taken over the natural numbers
$N(\mathbf{z}^{-}),N(\tilde{\mathbf{z}}^{-})$ and over the sets of (complex)
numbers

$\mathbf{z}^{\pm}=(z^{\pm}_{1},\dots,z^{\pm}_{N(\mathbf{z}^{\pm})}),\quad\mathbf{z}^{\pm}=(\tilde{z}^{\pm}_{1},\dots,z^{\pm}_{N(\tilde{\mathbf{z}}^{\pm})}),$
(1.21)

where
$N(\mathbf{z}^{+})+N(\mathbf{z}^{-})=2N,\,N(\tilde{\mathbf{z}}^{+})+N(\tilde{\mathbf{z}}^{-})=2\tilde{N}$,
and implies that $\mathbf{z}^{+}\cup\mathbf{z}^{-}=\mathbf{a}\cup\mathbf{b})$
and
$\tilde{\mathbf{z}}^{+}\cup\tilde{\mathbf{z}}^{-}=\tilde{\mathbf{a}}\cup\tilde{\mathbf{b}}$.
The coefficients denoted by square brackets will be written down in Section
2.6, see Theorem 4.1.

The simplest and trivial example of relation (1.20) is the case where
$\hat{g}_{\mu,\lambda}=\delta_{\mu,\lambda}$ (the case $\hat{g}=1$) and
$\mathbf{a}=\mathbf{b},\,\tilde{\mathbf{a}}=\tilde{\mathbf{b}}$. In the power
sum variables it is written as

$\sum_{\lambda}s_{\lambda}(\mathbf{p}^{\prime})s_{\lambda}(\tilde{\mathbf{p}}^{\prime})=e^{\sum_{n>0,\,{\rm
odd}}\frac{1}{n}p_{n}\tilde{p}_{n}}=\left(e^{\sum_{n>0}\frac{2}{n}p^{\rm
B}_{n}\tilde{p}_{n}^{\rm
B}}\right)^{2}=\left(\sum_{\mu}2^{-\ell(\mu)}Q_{\mu}({\mathbf{p}^{\rm
B}})Q_{\mu}(\tilde{\mathbf{p}}^{\rm B})\right)^{2}$ (1.22)

where the sets $\mathbf{p}^{\prime}$ and ${\mathbf{p}^{\rm B}}$ are the same
as in (1.4), (1.2) and (1.5). In the variables $\mathbf{a}$,
$\tilde{\mathbf{a}}$ (1.22) the both sides of (1.22) are equal to

$\left(\prod_{N\geq i>j}\frac{a_{i}-a_{j}}{a_{i}+a_{j}}\prod_{\tilde{N}\geq
i>j}\frac{\tilde{a}_{i}-\tilde{a}_{j}}{\tilde{a}_{i}+\tilde{a}_{j}}\right)^{2}.$
(1.23)

#### (d) Bi-lattice KP and bi-lattice BKP tau functions.

In view of the notion of the lattice tau functions (1.11) and (1.12) it is
natural to call $\hat{g}_{\lambda,\tilde{\lambda}}$ bi-lattice KP tau function
and to call $\hat{h}_{\mu,\tilde{\mu}}$ bi-lattice BKP tau function. Then we
get

$\hat{g}_{(\alpha|\beta),(\tilde{\alpha}|\tilde{\beta})}=\sum_{m(\zeta^{-}),m(\tilde{\zeta}^{-})\atop\zeta^{-},\zeta^{+},\tilde{\zeta}^{+},\tilde{\zeta}^{-}}\left[\zeta^{+},\zeta^{-}\atop\alpha,\beta\right]\left[\tilde{\zeta}^{+},\tilde{\zeta}^{-}\atop\tilde{\alpha},\tilde{\beta}\right]h^{+}_{\zeta^{+},\tilde{\zeta}^{+}}h^{-}_{\zeta^{-},\tilde{\zeta}^{-}}$
(1.24)

where $\zeta^{+}\cup\zeta^{-}=\alpha\cup(\beta+1)$ and
$\tilde{\zeta}^{+}\cup\tilde{\zeta}^{-}=\tilde{\alpha}\cup(\tilde{\beta}+1)$
and where $\lambda=(\alpha|\beta)$,
$\tilde{\lambda}=(\tilde{\alpha}|\tilde{\beta}))$ and
$\zeta^{\pm}=(\zeta^{\pm}_{1},\dots,\zeta^{\pm}_{m(\zeta^{\pm})})$ and
$\tilde{\zeta}^{\pm}=(\tilde{\zeta}^{\pm}_{1},\dots,\tilde{\zeta}^{\pm}_{m(\tilde{\zeta}^{\pm})})$
where $\alpha=(\alpha_{1},\dots,\alpha_{r})$,
$\beta=(\beta_{1},\dots,\beta_{r})$,
$\tilde{\alpha}=(\tilde{\alpha}_{1},\dots,\tilde{\alpha}_{\tilde{r}})$,
$\tilde{\beta}=(\tilde{\beta}_{1},\dots,\tilde{\beta}_{\tilde{r}})$ and where
$m(\zeta^{+})+m(\zeta^{-})=2r$,
$m(\tilde{\zeta}^{+})+m(\tilde{\zeta}^{-})=2\tilde{r}$. The weights denoted by
square brackets will be written below.

The simple nontrivial example of the bi-lattice KP tau function
$\hat{g}_{\lambda,\tilde{\lambda}}$ is the product
$s^{*}_{\mu}(\lambda)\dim\,\lambda$ where $s^{*}_{\mu}(\lambda)$ is the so-
called shifted Schur function introduced by Okounkov in [4] and $\dim\lambda$
is the number of standard tablauox of the shape $\lambda$, see [17]. The
shifted Schur functions were used in an approach to the representation theory
developed by G.Olshanski and A.Okounkov in [29].

The simple nontrivial example of the bi-lattice BKP tau function $h_{\mu,\nu}$
is the product $Q^{*}_{\mu}(\nu)\dim^{\rm B}\mu$ where $Q^{*}_{\mu}(\nu)$ is
the shifted projective Schur function introduced by Ivanov in [12] and
$\dim^{\rm B}\mu$ is the number of the shifted standard tablauex of the shape
$\mu$. Functions $Q^{*}_{\mu}(\nu)$ are of use in the study of spin Hurwitz
numbers [18].

The relation between shifted Schur functions and the shifted projective Schur
functions was written done in [30].

##  2 Preliminaries

Here we review known basic facts and certain previous results and introduce
notations.

###  2.1 Charged and neutral fermions and currents [6]

The fermionic creation and annihilation operators satisfy the anticommutation
relations:

$[\psi_{j},\psi_{k}]_{+}=[\psi^{\dagger}_{j},\psi^{\dagger}_{k}]_{+}=0,\quad[\psi_{j},\psi^{\dagger}_{k}]_{+}=\delta_{jk}.$
(2.1)

The vacuum element $|n\rangle$ in each charge sector ${\mathcal{F}}_{n}$ is
the basis element corresponding to the trivial partition $\lambda=\emptyset$:

$|n\rangle:=|\emptyset;n\rangle=e_{n-1}\wedge e_{n-2}\wedge\cdots.$ (2.2)

Elements of the dual space ${\mathcal{F}}^{*}$ are denoted as bra vectors
$\langle w|$, with the dual basis $\\{\langle\lambda;n|\\}$ for
${\mathcal{F}}^{*}_{n}$ defined by the pairing

$\langle\lambda;n|\mu;m\rangle=\delta_{\lambda\mu}\delta_{nm}.$ (2.3)

For KP $\tau$-functions, we need only consider the $n=0$ charge sector
${\mathcal{F}}_{0}$, and generally drop the charge $n$ symbol, denoting the
basis elements simply as

$|\lambda\rangle:=|\lambda;0\rangle.$ (2.4)

For $j>0$, $\psi_{-j}$ and $\psi^{\dagger}_{j-1}$ (resp. $\psi^{\dagger}_{-j}$
and $\psi_{j-1}$) annihilate the right (resp. left) vacua:

$\displaystyle\psi_{-j}|0\rangle$
$\displaystyle{\hskip-20.0pt}=0,\quad\psi^{\dagger}_{j-1}|0\rangle=0,\quad\forall
j>0,$ (2.5) $\displaystyle\langle 0|\psi^{\dagger}_{-j}$
$\displaystyle{\hskip-20.0pt}=0,\quad\langle 0|\psi_{j-1}=0,\quad\forall j>0.$
(2.6)

Neutral fermions $\phi^{+}_{j}$ and $\phi^{-}_{j}$ are defined [2] by

$\phi^{+}_{j}:=\frac{\psi_{j}+(-1)^{j}\psi^{\dagger}_{-j}}{\sqrt{2}},\quad\phi^{-}_{j}:=i\frac{\psi_{j}-(-1)^{j}\psi^{\dagger}_{-j}}{\sqrt{2}},\quad
j\in\mathbb{Z}$ (2.7)

(where $i=\sqrt{-1}$), and satisfy

$[\phi^{+}_{j},\phi^{-}_{k}]_{+}=0,\quad[\phi^{+}_{j},\phi^{+}_{k}]_{+}=[\phi^{-}_{j},\phi^{-}_{k}]_{+}=(-1)^{j}\delta_{j+k,0}.$
(2.8)

In particular,

$(\phi^{+}_{0})^{2}=(\phi^{-}_{0})^{2}=\tfrac{1}{2}.$ (2.9)

Acting on the vacua $|0\rangle$ and $|1\rangle$, we have

$\displaystyle\phi^{+}_{-j}|0\rangle$
$\displaystyle{\hskip-20.0pt}=\phi^{-}_{-j}|0\rangle=\phi^{+}_{-j}|1\rangle=\phi^{-}_{-j}|1\rangle=0,\quad\forall
j>0,\quad\forall j>0,$ (2.10) $\displaystyle\langle 0|\phi^{+}_{j}$
$\displaystyle{\hskip-20.0pt}=\langle 0|\phi^{-}_{j}=\langle
1|\phi^{+}_{j}=\langle 1|\phi^{-}_{j}=0,\quad\forall j>0,$ (2.11)
$\displaystyle\phi^{+}_{0}|0\rangle$
$\displaystyle{\hskip-20.0pt}=-i\phi^{-}_{0}|0\rangle=\tfrac{1}{\sqrt{2}}\psi_{0}|0\rangle=\tfrac{1}{\sqrt{2}}|1\rangle,$
(2.12) $\displaystyle\langle 0|\phi^{+}_{0}$
$\displaystyle{\hskip-20.0pt}=i\langle
0|\phi^{-}_{0}=\tfrac{1}{\sqrt{2}}\langle
0|\psi_{0}^{\dagger}=\tfrac{1}{\sqrt{2}}\langle 1|.$ (2.13)

#### Lie groups. Factorization condition.

Let us define the normal ordering $:\psi_{j}\psi_{k}:$ of $\psi_{j}\psi_{k}$
as $\psi_{j}\psi_{k}-\langle 0|\psi_{j}\psi_{k}|0\rangle$ and the normal
ordering of $\phi^{\pm}_{j}\phi^{\pm}_{k}$ as
$:\phi^{\pm}_{j}\phi^{\pm}_{k}:=\phi^{\pm}_{j}\phi^{\pm}_{k}-\langle
0|\phi^{\pm}_{j}\phi^{\pm}_{k}|0\rangle$.

Let us denote

$\hat{g}=e^{\sum_{j,k}A_{j,k}:\psi_{j}\psi^{\dagger}_{j}:}$ (2.14)

where $A_{j,k}$ are complex numbers.

If we ask $A_{j,k}$ decay in a fast enough way as $|j-k|\to\infty$, then the
exponents form Lie algebrs $\widehat{gl}_{\infty}$, and elements (2.14) form
${GL}_{\infty}$ group.

######  Remark 2.1.

“Such enough” implies the possibility of the exponentials to form Lie algebra,
where the commutator is well-defined. Say, finite matrices satisfy this. The
other example are the generalized Jacobian matrices (matrices with a finite
number of nonvanishing diagonals).

The elements

$\hat{h}^{\pm}(B)=e^{\sum_{j,k}B_{j,k}:\phi^{\pm}_{j}\phi^{\pm}_{j}:}$ (2.15)

where $B_{j,k}$ are complex numbers and $B_{j,k}=-B_{k,j}$.

If we ask $B_{j,k}$ decay in a fast enough way as $|j-k|\to\infty$, the
elements $\hat{h}^{\pm}(B)$ form $B_{\infty}$ group.

One can check the equality:

$(-1)^{k}:\psi_{j}\psi^{\dagger}_{-k}:-(-1)^{j}:\psi_{k}\psi^{\dagger}_{-j}:=:\phi^{+}_{j}\phi^{+}_{k}:+:\phi^{-}_{j}\phi^{-}_{k}:,$
(2.16)

Then we have the important

######  Lemma 2.1.

[6] Suppose

$\hat{g}=e^{\sum_{j,k}B_{jk}\left((-1)^{k}:\psi_{j}\psi^{\dagger}_{-k}:-(-1)^{j}:\psi_{k}\psi^{\dagger}_{-j}:\right)}$
(2.17)

where $B$ is a given (perhaps infinite) antisymmetric matrix. Then

$\hat{g}=\hat{h}^{+}\hat{h}^{-}$ (2.18)

where $\hat{h}^{\pm}$ are defined by (2.15):

$\hat{h}^{\pm}(B)=e^{\sum_{j,k}B_{j,k}:\phi^{\pm}_{j}\phi^{\pm}_{j}:}$ (2.19)

######  Remark 2.2.

The form (2.17) says that $\hat{g}\in B_{\infty}$.

#### Fermi fields.

Introduce

$\psi(z)=\sum_{i\in\mathbb{Z}}z^{i}\psi_{i},\quad\psi^{\dagger}(z)=\sum_{i\in\mathbb{Z}}z^{-i-1}\psi^{\dagger}_{i}$
(2.20)

and

$\phi^{\pm}(z)=\sum_{i\in\mathbb{Z}}z^{i}\phi^{\pm}_{i}.$ (2.21)

where $z$ is a formal parameter555In Eucledian 2D QFT $z=e^{i\varphi-\tau}$
where $\varphi$ has the meaning of the coordinate of the fermion $\psi(z)$ (of
$\phi(z)$) and $\tau$ has the meaning of time variable. . From (2.7) it
follows

$\psi(z)=\frac{\phi^{+}(z)-i\phi^{-}(z)}{\sqrt{2}},\quad\psi^{\dagger}(-z)=\frac{\phi^{+}(z)+i\phi^{-}(z)}{z\sqrt{2}}.$
(2.22)

######  Remark 2.3.

The formula

$\psi_{j}=\frac{\phi^{+}_{j}-i\phi^{-}_{j}}{\sqrt{2}},\quad(-1)^{j}\psi^{\dagger}_{-j}=\frac{\phi^{+}_{j}+i\phi^{-}_{j}}{\sqrt{2}}$
(2.23)

is quite similar to formula (2.22) and is rather similar to the relations

$\psi(-z^{-1})=\frac{\phi^{+}(-z^{-1})-i\phi^{-}(-z^{-1})}{\sqrt{2}},\quad\psi^{\dagger}(z^{-1})=-z\frac{\phi^{+}(-z^{-1})+i\phi^{-}(-z^{-1})}{\sqrt{2}}.$
(2.24)

These equations result in the usefull equalities:

$(-1)^{j}\psi_{j}\psi^{\dagger}_{-j}=i\phi^{+}_{j}\phi^{-}_{j}\,,$ (2.25)
$\psi(z)\psi^{\dagger}(-z)=\tfrac{i}{z}\phi^{+}(z)\phi^{-}(z)\,,$ (2.26)
$\psi(-z^{-1})\psi^{\dagger}(z^{-1})=-iz\phi^{+}(-z^{-1})\phi^{-}(-z^{-1})\,.$
(2.27)

We say that $\psi_{j},\psi^{\dagger}_{j}$ and $\phi^{\pm}_{j}$ are the Fermi
modes of the Fermi fields $\psi(z),\psi^{\dagger}_{j}(z)$ and $\phi^{\pm}(z)$
respectively.

#### Pairwise VEV.

From

$\langle 0|\psi_{k}\psi^{\dagger}_{j}|0\rangle=\begin{cases}\delta_{k,j},\quad
k<0,\\\ 0,\quad k>0.\end{cases}$ (2.28) $\langle
0|\phi^{\pm}_{k}\phi^{\pm}_{j}|0\rangle=\begin{cases}(-1)^{k}\delta_{k,-j},\quad
k<0,\\\ 0,\quad k>0.\end{cases}$ (2.29) $\langle
0|\phi^{\pm}_{k}\phi^{\mp}_{j}|0\rangle=\pm\frac{i}{2}\delta_{k,0}\delta_{j.0}$
(2.30)

one can easily obtain by direct calculation:

$\langle
0|\psi(a_{k})\psi^{\dagger}(-b_{j})|0\rangle=\frac{1}{b_{j}+a_{k}}\,,$ (2.31)
$\langle
0|\phi^{\pm}(a_{k})\phi^{\pm}(a_{j})|0\rangle=\frac{1}{2}\frac{a_{k}-a_{j}}{a_{k}+a_{j}}\,,$
(2.32) $\langle 0|\phi^{\pm}(a_{i})\phi^{\mp}(a_{j})|0\rangle=\pm\frac{i}{2}.$
(2.33)

######  Remark 2.4.

Actually the time order666In 2D QFT the argument of the Fermi field is written
as $x=e^{\sqrt{-1}\varphi-\tau}$, where $\varphi$ is the space and $\tau$ is
the time coordinate of the fermion. (the operation $\mathbb{T}$) is implied
for the pairwise correlation function (say, in case of neutral fermions):

$\langle
0|\mathbb{T}\left[\phi^{\pm}(z_{a})\phi^{\pm}(z_{b})\right]|0\rangle:=\begin{cases}\langle
0|\phi^{\pm}(z_{a})\phi^{\pm}(z_{b})|0\rangle\quad{\rm
if}\,|z_{a}|>|z_{b}|,\\\ -\langle
0|\phi^{\pm}(z_{b})\phi^{\pm}(z_{a})|0\rangle\quad{\rm
if}\,|z_{a}|<|z_{b}|.\end{cases}$ (2.34)

Say, for $|z_{a}|>|z_{b}|$ we have

$\langle 0|\phi^{\pm}(z_{a})\phi^{\pm}(z_{b})|0\rangle=\langle
0|\sum_{i}\phi^{\pm}_{i}(z_{a})^{i}\sum_{j}\phi^{\pm}_{j}(z_{b})^{j}|0\rangle=\frac{1}{2}+\sum_{j>0}(-1)^{j}\left(\frac{z_{b}}{z_{a}}\right)^{j}=\frac{1}{2}\frac{1-\frac{z_{b}}{z_{a}}}{1+\frac{z_{b}}{z_{a}}}.$
(2.35)

While in case $|z_{a}|<|z_{b}|$ we get
$-\frac{1}{2}\frac{1-\frac{z_{a}}{z_{b}}}{1+\frac{z_{a}}{z_{b}}}$. In both
cases ($|z_{a}|>|z_{b}|$ and $|z_{a}|<|z_{b}|$) the answer can be written as
$\frac{z_{a}-z_{b}}{z_{a}+z_{b}}$ (and the limit $|z_{a}|\to|z_{b}|$ where
$z_{a}\neq-z_{b}$ is smooth). We see, if our goal is to present
$\frac{1}{2}\frac{z_{a}-z_{b}}{z_{a}+z_{b}}$ as the correlation function, we
imply the time ordering. The same convention will be true for higher
correlation functions of neutral fermions. The point is that the final
formulas are clever enough not to take special care about the time ordering.

Next: the time ordering for the charged fermions:

$\langle
0|\mathbb{T}^{\prime}\left[\psi(x)\psi^{\dagger}(y)\right]|0\rangle:=\begin{cases}\langle
0|\psi(x)\psi^{\dagger}(y)|0\rangle\quad{\rm if}\,|x|>|y|,\\\ -\langle
0|\psi^{\dagger}(y)\psi(x)|0\rangle\quad{\rm if}\,|x|<|y|.\end{cases}$ (2.36)

In case $|x|=|y|$ and $x\neq y$ the fermions $\psi(x)$ and $\psi^{\dagger}(y)$
anticommute (Dirac delta function on the circle,
$\frac{dy}{y}\sum_{n\in\mathbb{Z}}\frac{a^{n}}{b^{n}}$, vanishes).

There is the following matching:

$\langle
0|\mathbb{T}^{\prime}\left[\psi(x)\psi^{\dagger}(-y)\right]|0\rangle=\langle
0|\mathbb{T}\left[\left(\frac{\phi^{+}(x)-i\phi^{-}(x)}{\sqrt{2}}\right)\left(\frac{\phi^{+}(y)+i\phi^{-}(y)}{y\sqrt{2}}\right)\right]|0\rangle.$
(2.37)

So $\mathbb{T}^{\prime}$-ordering is consistent with $\mathbb{T}$-ordering.
This is also correct for all higher correlators.

In what follows we shall not return to this topic and will never use
$\mathbb{T}$ symbols.

#### Factorization lemma.

As it was done in [9],[10],[11] we need Lemmas 3.2, 3.4, 3.47 to calculate tau
functions we need

######  Lemma 2.2 (Factorization).

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:

$\langle 0|U^{+}U^{-}|0\rangle=\begin{cases}\langle 0|U^{+}|0\rangle\langle
0|U^{-}|0\rangle&\text{ if }U^{+}\text{ and }U^{-}\text{ are both of even
degree}\\\ 0&\text{ if }U^{+}\text{ and }U^{-}\text{ have different parity
}\\\ 2i\langle 0|U^{+}\phi^{+}_{0}|0\rangle\langle
0|U^{-}\phi^{-}_{0}|0\rangle&\text{ if }U^{+}\text{ and }U^{+}\text{ are both
of odd degree}.\end{cases}$ (2.38)

#### Currents.

Define currents

$J_{m}=\sum_{i\in Z}\psi_{i}\psi^{\dagger}_{i+m},\quad m=\pm 1,\pm 2,\pm
3,\dots\,,$ (2.39) $J^{{\rm B}\pm}_{m}=\frac{1}{2}\sum_{i\in
Z}(-)^{i}\phi^{\pm}_{-i-m}\phi^{\pm}_{i},\quad m\,{\rm odd}.$ (2.40)

We have from (2.7)

$J_{m}=J^{{\rm B}+}_{m}+J^{{\rm B}-}_{m},\quad m\,{\rm odd},$ (2.41)
$J_{m}=\sqrt{-1}\sum_{j\in\mathbb{Z}}\phi^{+}_{j-m}\phi^{-}_{-j},\quad m\,{\rm
even}.$ (2.42)

The currents form the Heisenberg algebras as follows:

$[J_{m},J_{n}]=m\delta_{m,n},\quad m\neq 0,$ (2.43) $[J^{{\rm
B}\pm}_{m},J^{{\rm B}\pm}_{n}]=\frac{1}{2}m\delta_{m+n,0},\quad[J^{{\rm
B}+}_{m},J^{{\rm B}-}_{n}]=0,\quad m,n\,{\rm odd}.$ (2.44)

One can see that

$J_{m}|0\rangle=0=\langle 0|J_{-m},\quad m>0$ (2.45)

and

$J^{{\rm B}\pm}_{m}|0\rangle=0=\langle 0|J^{{\rm B}\pm}_{-m},\quad m>0$ (2.46)

###  2.2 Some notations we use (notational usage of partitions)

#### Partitions.

We recall that a nonincreasing set of nonnegative integers
$\lambda_{1}\geq\cdots\geq\lambda_{k}\geq 0$, we call partition
$\lambda=(\lambda_{1},\dots,\lambda_{l})$, and $\lambda_{i}$ are called parts
of $\lambda$. The sum of parts is called the weight $|\lambda|$ of $\lambda$.
The number of nonzero parts of $\lambda$ is called the length of $\lambda$, it
will be denoted $\ell(\lambda)$. See [17] for details. Partitions will be
denoted by Greek letters: $\lambda,\mu,\dots$. The set of all partitions is
denoted by $\mathop{\mathrm{missing}}{P}\nolimits$. The set of all partitions
with odd parts is denoted $\mathop{\mathrm{missing}}{OP}\nolimits$. Partitions
with distinct parts are called strict partitions, we prefer letters
$\alpha,\beta,\zeta$ (also $\zeta^{\pm}$) to denote them. The set of all
strict partitions will be denoted by $\mathop{\mathrm{missing}}{DP}\nolimits$.
The Frobenius coordinated $\alpha,\beta$ for partitions
$(\alpha|\beta)=\lambda\in\mathop{\mathrm{missing}}{P}\nolimits$ are of
usenames (let me recall that the coordinates
$\alpha=(\alpha_{1},\dots,\alpha_{r})\in\mathop{\mathrm{missing}}{DP}\nolimits$
consists of the lengths of arms counted from the main diagonal (the diagonal
nodes are not included) of the Young diagram of $\lambda$ while
$\beta=(\beta_{1},\dots,\beta_{r})\in\mathop{\mathrm{missing}}{DP}\nolimits$
consists of the lengths of legs counted from the main diagonal (again without
the nodes on the main diagonal) of the Young diagram of $\lambda$, $r$ is the
length of the main diaginal of $\lambda$, we call it Frobenius rank.

For example, the partition $(1)$ in Frobenius coordinated is written as
$(0|0)$ and it’s Frobenius rank is 1. Other examples: the partitions
$(1,1,1)$, $(n)$ and $(n,m),\,m\geq 2$ in the Frobenius coordinates are
written respectively as $(0|2)$ (rank equal to 1), $(n-1|0)$ (rank equal to 1)
and $(n-1,m-2|1,0))$ (rank equal to 2). See [17] for details about partitions
and their Young diagrams.

The number of the nonvanishing part of a partition $\lambda$ is called the
length of $\lambda$ and is denoted $\ell(\lambda)$. The sum of parts is called
the weight of $\lambda$ and denoted by $|\lambda|$. A partition also can be
presented as $\lambda=(1^{m_{1}}2^{m_{2}}3^{m_{3}}\cdots)$ where $m_{i}$ is
the number of time the number $i$ occurs in the partition. We shall use the
notation

$z_{\lambda}=\prod_{i}m_{i}!i^{m_{i}}.$ (2.47)

######  Definition 2.1.

Supplemented partitions.  If $\zeta$ is a strict partition of cardinality
$m(\zeta)$ (with $0$ allowed as a part), we define the associated supplemented
partition $\hat{\zeta}$ to be

$\hat{\zeta}:=\begin{cases}\zeta\ \text{ if }m(\zeta)\ \text{ is
even},\cr(\zeta,0)\ \text{ if }m(\zeta)\ \text{ is odd}.\end{cases}$ (2.48)

We denote by $m(\hat{\zeta})$ the cardinality of $\hat{\zeta}$ which is an
even number.

#### The ordered coordinate sets.

Consider given sets $\mathbf{a},\mathbf{b}$ of complex numbers

$\mathbf{a}=(a_{1},\dots,a_{N}),\quad\mathbf{b}=(b_{1},\dots,b_{N}),$ (2.49)

We want to have the similarity of the sets of complex numbers with partitions.
For this purpose let us introduce the following order in the set
$\mathbf{a}\cup\mathbf{b}$: in the set called ordered we place complex number
as a set with the weakly decaying set with respect to the absolute values of
the numbers. In case the absolute values in a subset is the same we just fix
any order inside the subset and keep it in what follows. Such sets we will
call time ordered sets777We recall that the complex coordinate, say $z$ of a
quantum field in Euclean 2D theory is presented as $e^{\sqrt{-1}\varphi-\tau}$
where $\tau$ is interpreted as time variable. Let us also treat the subsets
(LABEL:Just_sets) $\mathbf{a}$ and $\mathbf{b}$ of the set
$\mathbf{a}\cup\mathbf{b}$ as also ordered: $|a_{1}|\geq\cdots\geq|a_{N}|$ and
$|b_{1}|\geq\cdots\geq|b_{N}|$.

Starting from now all coordinate sets will be treated as the ordered sets
whose each member is labelled in the appropriate way (labeles goes up from the
left to the right). One can consider another pair of the ordered complementary
subsets, let us denote it as $\mathbf{z}^{+}$ and $\mathbf{z}^{-}$,
$\mathbf{z}^{+}\cup\mathbf{z}^{-}=\mathbf{a}\cup\mathbf{b}$ (the order
inherits the given order of the set $\mathbf{a}\cup\mathbf{b}$) whose
cardinalities $N(\mathbf{z}^{+})$ and $N(\mathbf{z}^{-})$ are not necessaryly
equal to $N$:

$\mathbf{z}^{\pm}=(z^{\pm}_{1},\dots,z^{\pm}_{N(\mathbf{z}^{\pm})}),\quad|z^{\pm}_{i}|\geq|z^{\pm}_{i+1}|,\quad
N(\mathbf{z}^{+})+N(\mathbf{z}^{-})=2N$ (2.50)

In what follow sets (2.49) and (2.50) will play different roles: the sets
(2.49) and the given number $N$ are constant along the paper while the sets
$\mathbf{z}^{+}$ and $\mathbf{z}^{-}$ will be varying and the numbers
$N(\mathbf{z}^{\pm})$ can be summation indices.

On the given sets (2.49) and also on sets (2.50) let us introduce the
following involution $*$

$a^{*}_{i}:=-a^{-1}_{N-i},\quad b^{*}_{i}:=-b^{-1}_{N-i},\quad
z^{\pm*}_{i}:=-\left(z^{\pm}_{N(\mathbf{z})-i}\right)^{-1},$ (2.51)

Then the sets

$\mathbf{a}^{*}=(a^{*}_{1},\dots,a^{*}_{N}),\quad\mathbf{b}^{*}=(b^{*}_{1},\dots,b^{*}_{N}),\quad\mathbf{z}^{\pm*}=(z^{\pm*}_{1},\dots,z^{\pm*}_{N(\mathbf{z}^{\pm})})$
(2.52)

are also ordered in the sense that $|a^{*}_{i}|\geq|a^{*}_{i+1}|$,
$|b^{*}_{i}|\geq|b^{*}_{i+1}|$, $|z^{\pm*}_{i}|\geq|z^{\pm*}_{i+1}|$ and the
set $\mathbf{a}^{*}\cup\mathbf{b}^{*}$ is ordered.

By analogy with Definition 2.1 we need

######  Definition 2.2.

Supplemented set of coordinates $\mathbf{z}^{\pm}$,$\mathbf{z}^{\pm*}$.  If
$\mathbf{z}^{\pm}$ is a strict partition of cardinality $N(\mathbf{z}^{\pm})$
(with $0$ allowed as a part), we define the associated supplemented set
$\hat{\mathbf{z}}^{\pm}$ to be

$\hat{\mathbf{z}}^{\pm}:=\begin{cases}\mathbf{z}^{\pm}\ \text{ if
}N(\mathbf{z}^{\pm})\ \text{ is even},\cr(\mathbf{z}^{\pm},0)\ \text{ if
}N(\mathbf{z}^{\pm})\ \text{ is
odd}\end{cases},\quad\hat{\mathbf{z}}^{\pm*}:=\begin{cases}\mathbf{z}^{\pm*}\
\text{ if }N(\mathbf{z}^{\pm})\ \text{ is even},\cr(\mathbf{z}^{\pm*},\infty)\
\text{ if }N(\mathbf{z}^{\pm*})\ \text{ is odd}.\end{cases}$ (2.53)

We denote by $N(\hat{\mathbf{z}}^{\pm})$ and $N(\hat{\mathbf{z}}^{\pm*})$ the
cardinality of respectively $\hat{\mathbf{z}}^{\pm}$ and
$\hat{\mathbf{z}}^{\pm*}$. We get
$N(\hat{\mathbf{z}}^{\pm})=N(\hat{\mathbf{z}}^{\pm*})$ is an even number.

#### Vandermond-like products.

For given sets

$\mathbf{a}=(a_{1},\dots,a_{N}),\quad\mathbf{z}^{\pm}=(z^{\pm}_{1},\dots,z^{\pm}_{N(\mathbf{z}^{\pm})})$
(2.54)

we use the following notations:

$\Delta(\mathbf{a}):=\prod_{i<j\leq N}(a_{i}-a_{j}),$ (2.55) $\Delta^{\rm
B}(\mathbf{z}^{\pm}):=\prod_{i<j\leq
N(\mathbf{z}^{\pm})}\frac{z^{\pm}_{i}-z^{\pm}_{j}}{z^{\pm}_{i}+z^{\pm}_{j}},$
(2.56)
$\Delta(\mathbf{a}/-\mathbf{b})=\frac{\Delta(\mathbf{a})\Delta(\mathbf{b})}{\prod_{i,j=1}^{N}(a_{i}+b_{j})}=\mathrm{det}\left((a_{i}+b_{j})^{-1}\right)_{i,j=1,\dots,N}\,.$
(2.57)

As we see

$\Delta^{\rm B}(\hat{\mathbf{z}}^{\pm})=\Delta^{\rm B}(\mathbf{z}^{\pm}).$
(2.58)

One veryfies that

$\Delta(\mathbf{a}^{*}):=\prod_{i<j\leq
N}\left(-a_{N-i}^{-1}+a_{N-j}^{-1}\right)=(-1)^{-\frac{1}{2}N(N-1)}\Delta(\mathbf{a})\prod_{i=1}^{N}a_{i}^{-(N-1)}$
(2.59)

and we obtain

$\displaystyle\Delta(\mathbf{b}^{*}/-\mathbf{a}^{*}):$ $\displaystyle=$
$\displaystyle(-1)^{N^{2}}\frac{\Delta(\mathbf{b}^{*})\Delta(\mathbf{a}^{*})}{\prod_{i,j=1}^{N}(a_{i}^{-1}+b_{j}^{-1})}$
(2.60)
$\displaystyle=\mathrm{det}\left((-a_{N-i}^{-1}-b_{N-j}^{-1})^{-1}\right)_{i,j=1,\dots,N}$
$\displaystyle=$
$\displaystyle(-1)^{N^{2}}\Delta(\mathbf{a}/-\mathbf{b})\prod_{i=1}^{N}a_{i}b_{i}$
(2.61)

and

$\Delta^{\rm B}(\mathbf{z}^{\pm*})=\Delta^{\rm
B}(\mathbf{z}^{\pm})=\Delta^{\rm B}(\hat{\mathbf{z}}^{\pm*})=\Delta^{\rm
B}(\hat{\mathbf{z}}^{\pm}).$ (2.62)

#### Partitions for products of Fermi modes.

For a given $\lambda=(\alpha|\beta)\in\mathop{\mathrm{missing}}{P}\nolimits$
let us use the following notations

$\displaystyle\Psi_{\lambda}=$ $\displaystyle:=$
$\displaystyle(-1)^{\sum_{j=1}^{r}\beta_{j}}(-1)^{\tfrac{1}{2}r(r-1)}\psi_{\alpha_{1}}\cdots\psi_{\alpha_{r}}\psi^{\dagger}_{-\beta_{1}-1}\cdots\psi^{\dagger}_{-\beta_{r}-1}\,,$
(2.63) $\displaystyle\Psi^{*}_{\lambda}$ $\displaystyle:=$
$\displaystyle(-1)^{\sum_{j=1}^{r}\beta_{j}}(-1)^{\tfrac{1}{2}r(r-1)}\psi_{-\beta_{r}-1}\cdots\psi_{-\beta_{1}-1}\psi^{\dagger}_{\alpha_{r}}\cdots\psi^{\dagger}_{\alpha_{1}}\,.$
(2.64)

Note that $|\lambda\rangle=\Psi_{\lambda}|0\rangle$, $\langle\lambda|=\langle
0|\Psi^{*}_{\lambda}$.

Let
$\alpha=(\alpha_{1},\dots,\alpha_{k})\in\mathop{\mathrm{missing}}{DP}\nolimits$
where $\alpha_{k}\geq 0$. We also use the notation $m(\alpha)$ for the number
$k$.

Introduce

$\displaystyle\Phi^{\pm}_{\alpha}$ $\displaystyle:=$ $\displaystyle
2^{\frac{k}{2}}\phi^{\pm}_{\alpha_{1}}\cdots\phi^{\pm}_{\alpha_{k}}\,,$ (2.65)
$\displaystyle\Phi^{\pm}_{-\alpha}$ $\displaystyle:=$
$\displaystyle(-1)^{\sum_{i=1}^{k}\alpha_{i}}2^{\frac{k}{2}}\phi^{\pm}_{-\alpha_{k}}\cdots\phi^{\pm}_{-\alpha_{1}}\,.$
(2.66)

Apart of the products $\Phi^{\pm}_{\alpha}$ the products
$\Phi^{\pm}_{\hat{\alpha}}$ will be of use where $\hat{\alpha}$ denotes the
supplemented partition, see Definition 2.1. Then we get

$\langle 0|\Phi^{\pm}_{-\alpha}\Phi^{\pm}_{\beta}|0\rangle=\langle
0|\Phi^{\pm}_{-\hat{\alpha}}\Phi^{\pm}_{\hat{\beta}}|0\rangle=2^{\ell(\alpha)}\delta_{\alpha,\beta}\,,$
(2.67) $\langle 0|\Phi^{\pm}_{-\alpha}\Phi^{\mp}_{\beta}|0\rangle=\langle
0|\Phi^{\pm}_{-\hat{\alpha}}\Phi^{\mp}_{\hat{\beta}}|0\rangle=\pm\frac{i}{2}\delta_{\alpha,0}\delta_{\beta,0}\,.$
(2.68)

In what follows sometimes we shall write $\phi_{x}$ and $\Phi_{x}$ instead of
$\phi^{+}_{x}$ and $\Phi^{+}_{x}$ ommiting superscripts $\pm$. We hope it will
not produce a confusion.

#### Notations for products of Fermi fields.

Next, we introduce

$\displaystyle\Psi(\mathbf{a}/-\mathbf{b}):$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{N}\psi(a_{i})\psi^{\dagger}(-b_{i})$ (2.69)
$\displaystyle=$
$\displaystyle(-1)^{\tfrac{1}{2}N(N-1)}\psi(a_{1})\cdots\psi(a_{N})\psi^{\dagger}(-b_{1})\cdots\psi^{\dagger}(-b_{N}),$
(2.70) $\displaystyle\Psi(\mathbf{b}^{*}/-\mathbf{a}^{*})$ $\displaystyle=$
$\displaystyle\prod_{i=1}^{N}\psi(-b_{i}^{-1})\psi^{\dagger}(a_{i}^{-1})$
(2.71) $\displaystyle=$
$\displaystyle(-1)^{\tfrac{1}{2}N(N-1)}\psi(-b_{N}^{-1})\cdots\psi(-b_{1}^{-1})\psi^{\dagger}(a_{N}^{-1})\cdots\psi^{\dagger}(a_{1}^{-1})$
(2.72)

and

$\displaystyle\Phi^{\pm}(\mathbf{z}\pm)$ $\displaystyle:=$ $\displaystyle
2^{-\frac{N(\mathbf{z}^{\pm})}{2}}\phi^{\pm}(z^{\pm}_{1})\cdots\phi^{\pm}(z^{\pm}_{N(\mathbf{z}^{\pm})}),$
(2.73) $\displaystyle\Phi^{\pm}(\mathbf{z}^{\pm*})$ $\displaystyle=$
$\displaystyle
2^{-\frac{N(\mathbf{z}^{\pm})}{2}}\phi^{\pm}(z^{\pm*}_{1})\cdots\phi^{\pm}(z^{\pm*}_{N(z^{\pm})})$
(2.74)

where each $z^{\pm*}_{i}$ is defined in (2.51).

Now consider VEV of these products. By (2.31) and by the Wick’s rule (see
Appendix B) we have for (2.69)

$\langle 0|\Psi(\mathbf{a}/-\mathbf{b})|0\rangle=\mathrm{det}\left(\langle
0|\psi(a_{i})\psi^{\dagger}(-b_{j})|0\rangle\right)_{i,j=1,\dots,N}$ (2.75)
$=\Delta(\mathbf{a}/-\mathbf{b}).$ (2.76)

Then thanks to (2.61) we get

$\langle
0|\Psi(\mathbf{b}^{*}/-\mathbf{a}^{*})|0\rangle=(-1)^{N^{2}}\Delta(\mathbf{a}/-\mathbf{b})\prod_{i=1}^{N}a_{i}b_{i}\,.$
(2.77)

Similarly from (2.32) by Wick’s rule we obtain

$\langle 0|\Phi^{\pm}(\hat{\mathbf{z}})|0\rangle=\mathrm{Pf}\left(\langle
0|\phi^{\pm}(z_{i})\phi^{\pm}(z_{j})|0\rangle\right)_{i,j=1,\dots,N(\hat{\mathbf{z}})}$
(2.78) $=2^{-\frac{1}{2}N(\hat{\mathbf{z}})}\Delta^{\rm B}(\mathbf{z}).$
(2.79)

Then it follows that

$\langle
0|\Phi^{\pm}(\hat{\mathbf{z}}^{*})|0\rangle=2^{-\frac{1}{2}N(\hat{\mathbf{z}})}\Delta^{\rm
B}(\mathbf{z}^{*})=2^{-\frac{1}{2}N(\hat{\mathbf{z}})}\Delta^{\rm
B}(\mathbf{z}).$ (2.80)

######  Remark 2.5.

One can make sure that

$\langle
0|\Psi(\mathbf{a}/-\mathbf{a})|0\rangle=\Delta(\mathbf{a}/-\mathbf{a})=\left(\Delta^{\rm
B}(\mathbf{a})\right)^{2}\prod_{j=1}^{N}(2a_{j})^{-N}=\langle
0|\Phi^{+}(\mathbf{a})\Phi^{-}(\mathbf{a})|0\rangle\prod_{j=1}^{N}(2a_{j})^{-N}.$
(2.81)

The equality (2.81) can be obtained in a different way as follows:

$\psi(a_{j})\psi^{\dagger}(-a_{j})=\frac{i}{a_{j}}\phi^{+}(a_{j})\phi^{-}(a_{j})$
(2.82)

one gets

$\Psi(\mathbf{a}/-\mathbf{a})=(-1)^{\frac{1}{2}N(N-1)}i^{N}\Phi^{+}(\mathbf{a})\Phi^{-}(\mathbf{a})\prod_{j=1}^{N}\frac{1}{a_{j}}.$
(2.83)

Then we obtain (2.81) from (2.69),(2.75) and (2.31),(2.32),(2.78).

Let us introduce

$\tilde{\Psi}(\mathbf{a}/-\mathbf{b})=\frac{\Psi(\mathbf{a}/-\mathbf{b})}{\langle
0|\Psi(\mathbf{a}/-\mathbf{b})|0\rangle}$ (2.84)

and

$\tilde{\Phi}^{\pm}(\hat{\mathbf{z}})=\frac{\Phi^{\pm}(\hat{\mathbf{z}})}{\langle
0|\Phi^{\pm}(\hat{\mathbf{z}})|0\rangle}.$ (2.85)

So we have

$\langle 0|\tilde{\Psi}(\mathbf{a}/-\mathbf{b})|0\rangle=\langle
0|\tilde{\Phi}(\hat{\mathbf{z}})|0\rangle=1.$ (2.86)

In what follows apart of the products $\Phi^{\pm}(\mathbf{z})$ the products
$\Phi^{\pm}(\hat{\mathbf{z}})$ will be of use where $\hat{\mathbf{z}}$ denotes
the supplemented coordinate set, see Definition 2.2.

#### Partitions for currents.

We introduce

$J_{\Delta}:=\prod_{i=1}^{\ell(\Delta)}J_{\Delta_{i}},\qquad
J_{-\Delta}:=\prod_{i=1}^{\ell(\Delta)}J_{-\Delta_{i}},\quad\Delta\in\mathop{\mathrm{missing}}{P}\nolimits$
(2.87)

and

$J^{{\rm B}\pm}_{\Delta}:=\prod_{i=1}^{\ell(\Delta)}J^{{\rm
B}\pm}_{\Delta_{i}},\qquad J^{{\rm
B}\pm}_{-\Delta}:=\prod_{i=1}^{\ell(\Delta)}J^{{\rm
B}\pm}_{-\Delta_{i}},\quad\Delta\in\mathop{\mathrm{missing}}{OP}\nolimits.$
(2.88)

###  2.3 Bosonization formulas

The nice part of the classical integrability worked out by Kyoto school is a
number of bosonization formulas.

Following [2], [3] we consider

$\hat{\gamma}(\mathbf{p})=e^{\sum_{m>0}\frac{1}{m}p_{m}J_{m}},\quad\hat{\gamma}^{\dagger}(\mathbf{p})=e^{\sum_{m>0}\frac{1}{m}p_{m}J_{-m}}$
(2.89)

and

$\hat{\gamma}^{{\rm B}\pm}(2{\mathbf{p}^{\rm B}})=e^{\sum_{m>0,{\rm
odd}}\frac{2}{m}p^{\rm B}_{m}J^{{\rm
B}\pm}_{-m}},\quad\hat{\gamma}^{{\dagger}{\rm B}\pm}(2{\mathbf{p}^{\rm
B}})=e^{\sum_{m>0,{\rm odd}}\frac{2}{m}p^{\rm B}_{m}J^{{\rm B}\pm}_{-m}}$
(2.90)

where $\mathbf{p}=(p_{1},p_{2},p_{3},\dots)$ and ${\mathbf{p}^{\rm B}}=(p^{\rm
B}_{1},p^{\rm B}_{3},p^{\rm B}_{5},\dots)$ a given sets of parameters.

For our purposes let us introduce the following notations

$\mathbf{p}(\mathbf{a}/-\mathbf{b})=\left(p_{1}(\mathbf{a}/-\mathbf{b}),p_{2}(\mathbf{a}/-\mathbf{b}),p_{3}(\mathbf{a}/-\mathbf{b}),\dots\right)$
(2.91)

where

$p_{m}(\mathbf{a}/-\mathbf{b})=\sum_{i=1}^{N}\left(a_{i}^{m}-(-b_{i})^{m}\right),$
(2.92)

and also

${\mathbf{p}^{\rm B}}(\mathbf{z})=\left(p^{\rm B}_{1}(\mathbf{z}),p^{\rm
B}_{3}(\mathbf{z}),p^{\rm B}_{5}(\mathbf{z}),\dots\right)$ (2.93)

where

$p^{\rm B}_{m}(\mathbf{z})=\sum_{i=1}^{N(\mathbf{z})}z_{i}^{m}.$ (2.94)

######  Remark 2.6.

Let us note that for each $m$ and for all possible
$\mathbf{a},\tilde{\mathbf{a}}$ we get
$p_{2m}(\mathbf{a}/-\mathbf{a})=0=\tilde{p}_{2m}(\tilde{\mathbf{a}}/-\tilde{\mathbf{a}})$.

The following bosonization formulas are well-known:

$\psi(a)\psi^{\dagger}(b)=\frac{1}{a-b}e^{\sum_{m>0}\frac{1}{m}\left(b^{-m}-a^{-m}\right)J_{-m}}e^{\sum_{m>0}\frac{1}{m}\left(a^{m}-b^{m}\right)J_{m}}$
(2.95)

and

$\phi(a)\phi(b)=\tfrac{1}{2}\frac{a-b}{a+b}e^{-\sum_{m>0,{\rm
odd}}\frac{2}{m}\left(b^{-m}+a^{-m}\right)J^{\rm B}_{-m}}e^{\sum_{m>0,{\rm
odd}}\frac{2}{m}\left(a^{m}+b^{m}\right)J^{\rm B}_{m}},$ (2.96)

therefore one can write

$\displaystyle\hat{\gamma}^{\dagger}(\mathbf{p}(\mathbf{b}^{*}/-\mathbf{a}^{*}))\hat{\gamma}(\mathbf{p}(\mathbf{a}/-\mathbf{b}))=\tilde{\Psi}^{*}(\mathbf{a}/-\mathbf{b}),$
(2.97)
$\displaystyle\hat{\gamma}^{\dagger}(\mathbf{p}(\mathbf{a}/-\mathbf{b}))\hat{\gamma}(\mathbf{p}(\mathbf{b}^{*}/-\mathbf{a}^{*}))=\tilde{\Psi}(\mathbf{a}/-\mathbf{b}),$
(2.98) $\hat{\gamma}^{{\dagger}{\rm B}\pm}(2{\mathbf{p}^{\rm
B}}(\mathbf{z}^{*}))\hat{\gamma}^{{\rm B}_{\pm}}(2{\mathbf{p}^{\rm
B}}(\mathbf{z}))=\tilde{\Phi}^{*\pm}(\hat{\mathbf{z}}),\quad\hat{\gamma}^{{\dagger}{\rm
B}\pm}(2{\mathbf{p}^{\rm B}}(\mathbf{z}))\hat{\gamma}^{{\rm
B}_{\pm}}(2{\mathbf{p}^{\rm
B}}(\mathbf{z}^{*}))=\tilde{\Phi}^{\pm}(\hat{\mathbf{z}}).$ (2.99)

These relations are in agreement with (2.86) because of (2.45) and (2.46).

From above and from (2.45),(2.46) we get

$\langle 0|\hat{\gamma}(\mathbf{p}(\mathbf{a}/-\mathbf{b}))=\langle
0|\tilde{\Psi}(\mathbf{b}^{*}/-\mathbf{a}^{*}),\quad\hat{\gamma}^{\dagger}(\mathbf{p}(\mathbf{a}/-\mathbf{b}))|0\rangle=\tilde{\Psi}(\mathbf{a}/-\mathbf{b})|0\rangle$
(2.100)

and

$\langle 0|\hat{\gamma}^{{\rm B}_{\pm}}(2{\mathbf{p}^{\rm
B}}(\mathbf{z}))=\langle
0|\tilde{\Phi}^{\pm}(\hat{\mathbf{z}}^{*}),\quad\hat{\gamma}^{{\dagger}{\rm
B}\pm}(2{\mathbf{p}^{\rm
B}}(\mathbf{z}))|0\rangle=\tilde{\Phi}^{\pm}(\hat{\mathbf{z}})|0\rangle.$
(2.101)

###  2.4 Fermions and symmetric functions

The bosonization formulas are also manifested by the representation of the
known symmetric polynomials as fermionic vacuum expectation values. This is an
interesting part of the soliton theory.

#### Power sums and Schur functions.

For preliminary information about symmetric functions we recommend the
textbook [17]. This is about polynomial functions symmetric in variables
$a_{1},\dots,a_{N}$, the widely known examples are the so-called power sums
(or, the same, Newton sums) labels by a multiindex
$\lambda=(\lambda_{1},\dots,\lambda_{n})\in\mathop{\mathrm{missing}}{P}\nolimits$:

$\mathbf{p}_{\lambda}(\mathbf{a}):=p_{\lambda_{1}}p_{\lambda_{2}}\cdots
p_{\lambda_{n}},\quad{\rm where}\,\,\,p_{m}=\sum_{i=1}^{N}a_{i}^{m}$ (2.102)

or the Schur functions labeled by
$\lambda=(\lambda_{1},\dots,\lambda_{n})\in\mathop{\mathrm{missing}}{P}\nolimits$:

$s_{\lambda}(\mathbf{a})=\frac{\mathrm{det}\left(a_{i}^{\lambda_{j}-j+N}\right)_{i,j\leq
N}}{\Delta(\mathbf{a})}$ (2.103)

(where it is supposed that $n\leq N$). For the empty partition we put
$\mathbf{p}_{0}(\mathbf{a})=0$ and $s_{0}(\mathbf{a})=1$. The power sums and
the Schur functions are related:

$\mathbf{p}_{\lambda}=\sum_{\mu\atop|\mu|=|\lambda|}\chi_{\mu}(\lambda)s_{\mu}$
(2.104)

where the coefficients $\chi_{\mu}(\lambda)$ are very important in many
problems. $\chi_{\mu}(\lambda)$ has the meaning of the character of the
irreducible representation $\mu$ of the permutation group $S_{d}$, $d=|\mu|$
evaluated on the cycle class $\lambda$, see for instance the textbook [17].

On the space of polynomial symmetric functions denoted by $\Lambda_{N}$ the
scalar product is given by

$<\mathbf{p}_{\lambda},\mathbf{p}_{\mu}>=z_{\lambda}\delta_{\lambda,\mu}=\langle
0|J_{-\lambda}J_{\mu}|0\rangle,\quad\lambda,\mu\in\mathop{\mathrm{missing}}{P}\nolimits$
(2.105)

where $z_{\lambda}$ was defined in (2.47). Notice that the relation does not
include any $N$-dependence. The Schur functions form the orthonormal basis
there:

$<s_{\lambda},s_{\mu}>=\delta_{\lambda,\mu}=\langle
0|\Psi^{*}_{\lambda}\Psi_{\mu}|0\rangle,\quad\lambda,\mu\in\mathop{\mathrm{missing}}{P}\nolimits.$
(2.106)

The last equalities in both formulas (2.105) and (2.106) is the result of the
direct computation (using resectively (2.43),(2.45) and (2.1),(2.5)), however
it is not an occasion: it is the manifistation of the boson-fermion
correspondence which is very popular in 2D physics.

In case power where the sum variables are labeled with only odd parts, namely
if
$\lambda=(\lambda_{1},\dots,\lambda_{k})\in\mathop{\mathrm{missing}}{OP}\nolimits$,
we denote $\mathbf{p}_{\lambda}$ by $\mathbf{p}^{\rm
B}_{\lambda}=\prod_{i=1}^{\ell(\lambda)}p_{\lambda_{i}}^{\rm B}$. These
symmetric functions form the subspace of $\Lambda_{N}$ denoted by $\Gamma$ in
[17] (see Part III there). There is the following natural scalar product on
$\Gamma$:

$<\mathbf{p}^{\rm B}_{\lambda},\mathbf{p}^{\rm B}_{\mu}>_{\rm
B}=2^{-\ell(\lambda)}z_{\lambda}\delta_{\lambda,\mu}=\langle 0|J^{\rm
B}_{-\lambda}J^{\rm
B}_{\mu}|0\rangle,\quad\lambda,\mu\in\mathop{\mathrm{missing}}{OP}\nolimits.$
(2.107)

Also

$<Q_{\alpha},Q_{\beta}>_{\rm
B}=2^{\ell{(\alpha)}}\delta_{\alpha,\beta}=\langle
0|\Phi_{-\alpha}\Phi_{\beta}|0\rangle=\langle
0|\Phi_{-\hat{\alpha}}\Phi_{\hat{\beta}}|0\rangle,\quad\alpha,\beta\in\mathop{\mathrm{missing}}{DP}\nolimits,$
(2.108)

see [17]. The validity of the second equalities in (2.107) and in (2.108) is
obtained by the direct evaluation with the help respectively of (2.44),(2.46)
and of (2.8),(2.10),(2.12)).

We also have

$s_{\lambda}({\bf J})=\Psi_{\lambda},\quad s_{\lambda}({\bf
J}^{\dagger})=\Psi^{*}_{\lambda},\quad\lambda=(\alpha|\beta)\in\mathop{\mathrm{missing}}{P}\nolimits$
(2.109)

and

$Q_{\mu}({\bf J}^{{\rm B}\pm})=\Phi_{\hat{\mu}},\quad Q_{\mu}({\bf
J}^{{\dagger}{\rm
B}\pm})=\Phi_{-\hat{\mu}},\quad\mu\in\mathop{\mathrm{missing}}{DP}\nolimits$
(2.110)

where both Schur functions are considered as polynomials in power sum
variables and the role of power sums play respectively ${\bf
J}=(J_{1},J_{2},J_{3},\dots)$, ${\bf
J}^{\dagger}=(J^{\dagger}_{1},J^{\dagger}_{2},J^{\dagger}_{3},\dots)$ and
${\bf J}^{{\rm B}\pm}=(J^{{\rm B}\pm}_{1},J^{{\rm B}\pm}_{3},J^{{\rm
B}\pm}_{5},\dots)$, ${\bf J}^{{\dagger}{\rm B}\pm}=(J^{{\dagger}{\rm
B}\pm}_{1},J^{{\dagger}{\rm B}\pm}_{3},J^{{\dagger}{\rm B}\pm}_{5},\dots)$
(compare to [31]).

We get

$\langle
0|\hat{\gamma}(\mathbf{p})\hat{\gamma}^{\dagger}(\tilde{\mathbf{p}})|0\rangle=e^{\sum_{m>0}\frac{1}{m}p_{m}\tilde{p}_{m}}=\sum_{\lambda\in\mathop{\mathrm{missing}}{P}\nolimits}\frac{1}{z_{\lambda}}\mathbf{p}_{\lambda}\tilde{\mathbf{p}}_{\lambda}$
(2.111)
$=\sum_{\lambda\in\mathop{\mathrm{missing}}{P}\nolimits}s_{\lambda}(\mathbf{p})s_{\lambda}(\tilde{\mathbf{p}})$
(2.112)

which can be derived either from (2.43) or also from (2.105).

From (2.44)

$\langle 0|\hat{\gamma}^{{\rm B}\pm}(2{\mathbf{p}^{\rm
B}})\hat{\gamma}^{{\dagger}{\rm B}\pm}(2\tilde{\mathbf{p}}^{\rm
B})|0\rangle=e^{\sum_{m>0,{\rm
odd}}\frac{2}{m}p_{m}\tilde{p}_{m}}=\sum_{\lambda\in\mathop{\mathrm{missing}}{OP}\nolimits}2^{\ell(\lambda)}\frac{1}{z_{\lambda}}\mathbf{p}^{\rm
B}_{\lambda}\tilde{\mathbf{p}}^{\rm B}_{\lambda}$ (2.113)
$=\sum_{\mu\in\mathop{\mathrm{missing}}{DP}\nolimits}2^{-\ell(\mu)}Q_{\mu}({\mathbf{p}^{\rm
B}})Q_{\mu}(\tilde{\mathbf{p}}^{\rm B}).$ (2.114)

Let us write down the following equalities

$\langle
0|\hat{\gamma}(\mathbf{p})=\sum_{\lambda\in\mathop{\mathrm{missing}}{P}\nolimits}s_{\lambda}(\mathbf{p})\langle
0|\Psi^{\dagger}_{\lambda}\,\,,\quad\hat{\gamma}^{\dagger}(\tilde{\mathbf{p}})|0\rangle=\sum_{\lambda\in\mathop{\mathrm{missing}}{P}\nolimits}\Psi_{\lambda}|0\rangle
s_{\lambda}(\tilde{\mathbf{p}})$ (2.115)

which, thanks to (2.67), is equivalent to the Sato formula (2.132) below, and

$\langle 0|\hat{\gamma}^{{\rm B}}({\mathbf{p}^{\rm
B}})=\sum_{\mu\in\mathop{\mathrm{missing}}{DP}\nolimits}2^{-\ell(\mu)}Q_{\mu}({\mathbf{p}^{\rm
B}})\langle 0|\Phi_{-\mu}\,\,,\quad\hat{\gamma}^{{\dagger}{\rm
B}}(\tilde{\mathbf{p}}^{\rm
B})|0\rangle=\sum_{\mu\in\mathop{\mathrm{missing}}{DP}\nolimits}2^{-\ell(\mu)}\Phi_{\mu}|0\rangle
Q_{\mu}(\tilde{\mathbf{p}}^{\rm B}).$ (2.116)

which, thanks to (2.68), is equivalent to the relation found in [37], see
(2.133) below.

###  2.5 Fermions and tau functions: KP and BKP cases.

There are different treatments of the notion of tay function. Here we use the
fermionic approach to tau functions.

####  $\tau$ functions as vacuum expectation values (VEVs).

888We give one of possible definitions of the tau function. Here we want to
avoid the notions of the Plucker and the Cartan coordinates on Sato
grassmannian and isotropic grassmannian which are out of real use in the
present work.

According to [6] KP tau functions can be presented in form of the following
vacuum expectation value (VEV)

$\tau(\mathbf{p}|\hat{g})=\langle 0|\hat{\gamma}(\mathbf{p})\hat{g}|0\rangle$
(2.117)

where $\hat{g}$ is an exponential of a bilinear in $\\{\psi_{i}\\}$ and
$\\{\psi^{\dagger}_{i}\\}$ expression

$\hat{g}=e^{\sum_{i,j}A_{ij}\psi_{i}\psi^{\dagger}_{j}}$ (2.118)

where $A$ is an infinite matrix which is treated as
$A\in\operatorname{gl}(\infty)$ (see Section 2.1) and where
$\hat{\gamma}(\mathbf{p})$ is given by (2.89).

######  Remark 2.7.

Throughout the text we assume that the so-called matrix elements:

$\hat{g}_{\mu,\lambda}:=\langle
0|\Psi^{\dagger}_{\mu}\hat{g}\Psi_{\lambda}|0\rangle$ (2.119)

do exist for each pair $\mu,\lambda\in\mathop{\mathrm{missing}}{P}\nolimits$.
This assumption allows to identify tau functions with their Taylor series in
power sums by substituting (2.115) and (2.119) into (2.117).

The tau function depends on the choice of $\hat{g}$ (the same: on the choice
of the matrix $A$) and on the infinite set of parameters
$\mathbf{p}=(p_{1},p_{2},p_{3},\dots)$ (the power sum variables).

######  Remark 2.8.

Tau function (2.117) can be also written as

$\tau(\mathbf{p}|\hat{g})=\langle
0|\hat{g}^{\dagger}\hat{\gamma}^{\dagger}(\mathbf{p})|0\rangle$ (2.120)

where

$\hat{g}^{\dagger}=e^{\sum_{i,j}A_{ij}\psi_{j}\psi^{\dagger}_{i}}$ (2.121)

and $\hat{\gamma}^{\dagger}(\mathbf{p})$ is given by (2.89).

The BKP tau function depends on the set of odd-labeled power sums
${\mathbf{p}^{\rm B}}=(p_{1},p_{3},\dots)$ and can be presented as

$\tau^{B}(2{\mathbf{p}^{\rm B}}|\hat{h}^{\pm})=\langle 0|\hat{\gamma}^{\rm
B\pm}(2{\mathbf{p}^{\rm B}})\hat{h}^{\pm}|0\rangle$ (2.122)

where $\hat{h}^{\pm}$ is an exponential of a quadratic in
$\\{\phi^{\pm}_{i}\\}$ expression

$\hat{h}^{\pm}=e^{\sum_{i,j}B_{ij}\phi^{\pm}_{i}\phi^{\pm}_{j}}$ (2.123)

where ${B}^{\pm}$ is infinite antisymmetric matrices and where
$\hat{\gamma}^{\rm B\pm}$ is given by (2.90).

######  Remark 2.9.

We assume that the matrix elements:

$\hat{h}^{\pm}_{\mu,\nu}:=\langle
0|\Phi^{\pm}_{-\mu}\hat{h}\Phi_{\nu}|0\rangle$ (2.124)

do exist for each pair $\mu,\nu\in\mathop{\mathrm{missing}}{DP}\nolimits$.

######  Remark 2.10.

The set of $t_{m}=\frac{1}{m}p_{m},\,m=1,2,3,\dots$ is called the set of the
KP higher times and the set of $t_{m}^{\rm B}=\frac{2}{m}p_{m}^{\rm
B},\,m=1,3,5,\dots$ is called the set of the BKP higher times.

######  Remark 2.11.

Tau function (2.122) can be also written as

$\tau^{\rm B}({\mathbf{p}^{\rm B}}|\hat{h}^{\pm})=\langle
0|(\hat{h}^{\pm})^{\dagger}\hat{\gamma}^{{\dagger}{\rm B}\pm}({\mathbf{p}^{\rm
B}})|0\rangle$ (2.125)

where

$(\hat{h}^{\pm})^{\dagger}=e^{\sum_{i,j}B_{ij}\phi^{\pm}_{j}\phi^{\pm}_{i}}$
(2.126)

and $\hat{\gamma}^{{\dagger}{\rm B}\pm}({\mathbf{p}^{\rm B}})$ is given by
(2.90).

In the basic works of Kyoto school (for instance in [6]) the following
relation was proved

######  Theorem 2.3.

[3] Under conditions

$\hat{g}=\hat{h}^{+}\hat{h}^{-}$ (2.127)

(as written down in Lemma 2.1) and also under

${\mathbf{p}^{\rm B}}=(p^{\rm B}_{1},p^{\rm B}_{3},p^{\rm
B}_{5},\dots),\quad\mathbf{p}^{\prime}:=(p_{1},0,p_{3},0,p_{5},0,\dots),$
(2.128)

which are related by (1.2),(1.4),(1.5), the relation between KP tau function
(2.117) and (any the both) BKP tau functions (2.122) is as follows:

$\left(\tau^{\rm B}({\mathbf{p}^{\rm
B}}|\hat{h}^{\pm})\right)^{2}=\tau(\mathbf{p}^{\prime}|\hat{h}^{+}\hat{h}^{-})$
(2.129)

#### The Schur and the projective Schur functions as tau functions - a remark.

######  Remark 2.12.

For any given $\lambda=(\alpha|\beta)\in\mathop{\mathrm{missing}}{P}\nolimits$
and $\alpha\in\mathop{\mathrm{missing}}{DP}\nolimits$ there exists such
$\hat{g}=\hat{g}(\lambda)$ and such $\hat{h}^{\pm}=\hat{h}^{\pm}(\alpha)$ that

$\hat{g}(\lambda)|0\rangle=\Psi_{\lambda}|0\rangle,$ (2.130)
$\hat{h}^{\pm}(\alpha)|0\rangle=\Phi^{\pm}_{\hat{\alpha}}|0\rangle.$ (2.131)

This is the basic fact of the Sato theory. For a given
$\lambda=(\alpha|\beta)$ and $\alpha$ one can present both $\hat{g}(\lambda)$
and $\hat{h}^{\pm}(\alpha)$ in the explicit way, see Appendix.

The wonderful observation by Sato and his school [33], [6] is the fermionic
formula for the Schur polynomial

$s_{\lambda}(\mathbf{p})=\langle
0|\hat{\gamma}(\mathbf{p})\Psi_{\lambda}|0\rangle=\langle
0|\Psi^{*}_{\lambda}\hat{\gamma}^{\dagger}(\mathbf{p})|0\rangle.$ (2.132)

In the BKP case the similar formula was found in [37]:

$Q_{\alpha}({\mathbf{p}^{\rm B}})=\langle 0|\hat{\gamma}^{{\rm
B}\pm}(2{\mathbf{p}^{\rm B}}))\Phi^{\pm}_{\hat{\alpha}}|0\rangle=\langle
0|\Phi^{\pm}_{-\hat{\alpha}}\hat{\gamma}^{{\dagger}B\pm}(2{\mathbf{p}^{\rm
B}})|0\rangle.$ (2.133)

#### Schur functions $s_{\lambda}(\mathbf{a}/-\mathbf{b})$ and
$Q_{\mu}(\mathbf{z})$ and products of Fermi fields
$\Psi(\mathbf{a}/-\mathbf{b})$, $\Phi(\mathbf{z})$.

With the help of (2.100) and (2.101) we re-write formulas (2.132) and (2.133)
for the both Schur functions as follows:

$s_{\lambda}(\mathbf{p}(\mathbf{a}/-\mathbf{b}))=\langle
0|\tilde{\Psi}(\mathbf{b}^{*}/-\mathbf{a}^{*})\Psi_{\lambda}|0\rangle=\langle
0|\Psi^{*}_{\lambda}\tilde{\Psi}(\mathbf{a}/-\mathbf{b})|0\rangle=:s_{\lambda}(\mathbf{a}/-\mathbf{b})$
(2.134)

where $\lambda=(\alpha|\beta)$ and

$Q_{\alpha}({\mathbf{p}^{\rm B}}(\mathbf{z}))=\langle
0|\tilde{\Phi}(\hat{\mathbf{z}}^{*})\Phi_{\hat{\alpha}}|0\rangle=\langle
0|\Phi_{-\hat{\alpha}}\tilde{\Phi}(\hat{\mathbf{z}})|0\rangle=:Q_{\alpha}(\mathbf{z}).$
(2.135)

######  Remark 2.13.

By the limiting procedure we obtain from (2.134):

$s_{\lambda}(\mathbf{a})\Delta(\mathbf{a})=\langle-N|\psi^{\dagger}(a_{1}^{-1})\cdots\psi^{\dagger}(a_{N}^{-1})\Psi_{\lambda}|0\rangle=\langle
0|\Psi^{*}_{\lambda}\psi(a_{1})\cdots\psi(a_{N})|N\rangle$ (2.136)

where to get the first equality we send $b_{1}>\cdots>b_{N}\to\infty$ in the
second member of (2.134) and to get the second equality in (2.136)we send
$b_{N}<\cdots<b_{1}\to 0$ in the last member of (2.134).

we rewrite (2.134) as

$s_{\lambda}(\mathbf{a}/-\mathbf{b})\Delta(\mathbf{a}/-\mathbf{b})=\langle
0|\Psi(\mathbf{b}^{*}/-\mathbf{a}^{*})\Psi_{\lambda}|0\rangle=(-1)^{N^{2}}\langle
0|\Psi^{*}_{\lambda}\Psi(\mathbf{a}/-\mathbf{b})|0\rangle\prod_{i=1}^{N}a_{i}^{-1}b_{i}^{-1}.$
(2.137)

###  2.6 Two-sided KP tau function, two-sided BKP tau functions, lattice KP
tau function, lattice BKP tau function

All objects which we need are tau functions introduced in works of Kyoto
school. To be more precize in the terminology concerning tau functions which
we use let us give definitions.

#### Two-sided KP tau functions. Lattice KP tau functions.

######  Definition 2.3.

We call

$\tau(\mathbf{p},\tilde{\mathbf{p}}|\hat{g})=\langle
0|\,\hat{\gamma}(\mathbf{p})\,\hat{g}\,\hat{\gamma}(\tilde{\mathbf{p}})\,|0\rangle$
(2.138)

the two-sided KP (in short 2-KP) tau function. Here
$\mathbf{p}=(p_{1},p_{2},p_{3},\dots)$ and
$\tilde{\mathbf{p}}=(\tilde{p}_{1},\tilde{p}_{2},\tilde{p}_{3},\dots)$ are
parameters. We call $t_{m}=\frac{1}{m}p_{m}$ and
$\tilde{t}_{m}=\frac{1}{m}\tilde{p}_{m}$ the 2-KP higher times.

This notion of the KP higher times is the same as it was introduced in [3],
[24] and is common. Two-sided KP tau function is usual (one-sided) KP tau
function (2.117) with respect to the higher times $\mathbf{t}=\mathbf{p}$ and
at the same time it is one-sided KP tau function with respect to the higher
time variables $\tilde{\mathbf{t}}=\tilde{\mathbf{p}}$, see Remark 2.8 in
Section 2.5.

######  Definition 2.4.

We call

$S_{\lambda}(\mathbf{p}|\hat{g}):=\langle
0|\hat{\gamma}(\mathbf{p})\hat{g}\Psi_{\lambda}|0\rangle$ (2.139)

lattice KP tau function labelled with a partition
$\lambda\in\mathop{\mathrm{missing}}{P}\nolimits$.

With the assumption of Remark 2.7 one can write both tau function as Taylor
series in power sum variables as follows:

$\tau(\mathbf{p},\tilde{\mathbf{p}}|\hat{g})=\sum_{\mu,\lambda\in\mathop{\mathrm{missing}}{P}\nolimits}\hat{g}_{\mu,\lambda}s_{\mu}(\mathbf{p})s_{\lambda}(\tilde{\mathbf{p}})$
(2.140)
$S_{\lambda}(\mathbf{p}|\hat{g})=\sum_{\mu\in\mathop{\mathrm{missing}}{P}\nolimits}\hat{g}_{\mu,\lambda}s_{\mu}(\mathbf{p}))$
(2.141)

which results from (2.115). The series (2.140) was written down in [35], [36]
in the context of the study of the Toda lattice hierarchy.

For a given $\hat{g}$, the relation between two-sided and lattice KP tau
functions is as follows:

$\tau(\mathbf{p},\tilde{\mathbf{p}}|\hat{g})=\sum_{\lambda\in\mathop{\mathrm{missing}}{P}\nolimits}S_{\lambda}(\mathbf{p}|\hat{g})s_{\lambda}(\tilde{\mathbf{p}})$
(2.142)

In case $\hat{g}=1$ the last equality yields (see (2.111)-(2.112))

$\tau(\mathbf{p},\tilde{\mathbf{p}}|\hat{g}=1)=\sum_{\lambda\in\mathop{\mathrm{missing}}{P}\nolimits}s_{\lambda}(\mathbf{p})s_{\lambda}(\tilde{\mathbf{p}})=e^{\sum_{m>0}\frac{1}{m}p_{m}\tilde{p}_{m}}$
(2.143)

which may be called 2-KP vacuum tau function and occurs to be the simplest
example of 2-KP tau functions.

######  Remark 2.14.

In the Grassmannian approach to the tau functions [33] the coefficients
$S_{\lambda}(\mathbf{p}|\hat{g})$ in (2.142) have the meaning of the Plucker
coordinates $\pi_{\lambda}(\hat{g})$ for the one-side KP tau function, see
Remark 2.8 where one should insert $\tilde{\mathbf{p}}$ instead of
$\mathbf{p}$. The notation was used in [10] instead of
$S_{\lambda}(\mathbf{p}|\hat{g})$, see Remark 1.5.

#### Two-sided BKP tau functions. Lattice BKP tau functions.

######  Definition 2.5.

We call

$\tau(2{\mathbf{p}^{\rm B}},2\tilde{\mathbf{p}}^{\rm B}|\hat{h}^{\pm})=\langle
0|\,\hat{\gamma}^{\rm B\pm}(2{\mathbf{p}^{\rm
B}})\,\hat{h}^{\pm}\,\hat{\gamma}^{\rm B\pm}(2\tilde{\mathbf{p}}^{\rm
B})\,|0\rangle$ (2.144)

the two-sided BKP (2-BKP) tau function. Here ${\mathbf{p}^{\rm
B}}=(p_{1},p_{3},p_{5},\dots)$ and $\tilde{\mathbf{p}}^{\rm
B}=(\tilde{p}_{1},\tilde{p}_{3},\tilde{p}_{5},\dots)$ are parameters. We call
$t^{\rm B}_{m}=\frac{2}{m}p^{\rm B}_{m}$ and $\tilde{t}^{\rm
B}_{m}=\frac{2}{m}\tilde{p}^{\rm B}_{m}$ the two-sided BKP higher times.

This notion of the BKP higher times is the same as it was introduced in
[2],[3] and in [14] and is common.

Two-sided BKP tau function is usual (one-sided) BKP tau function (2.122) with
respect to the higher times $\mathbf{t}^{\rm B}=(t^{\rm B}_{1},t^{\rm
B}_{3},\dots)$ and at the same time it is one-sided KP tau function with
respect to the higher time variables $\tilde{\mathbf{t}}^{\rm
B}=(\tilde{t}^{\rm B}_{1},\tilde{t}^{\rm B}_{1},\dots)$, see Remark 2.11 in
Section 2.5.

######  Remark 2.15.

The case $\tilde{N}=0$, see (1.17), (1.19) is related to the (one-side) tau
function.

######  Definition 2.6.

We call

$K_{\mu}({\mathbf{p}^{\rm B}}|\hat{h}^{\pm}):=\langle 0|\hat{\gamma}^{{\rm
B}\pm}({\mathbf{p}^{\rm B}})\hat{h}^{\pm}\Phi_{\mu}|0\rangle,$ (2.145)

lattice BKP tau function labelled with a partition
$\mu\in\mathop{\mathrm{missing}}{DP}\nolimits$.

With the assumption of Remark 2.9 one can write both BKP tau functions as
Taylor series in power sum variables as follows:

$\tau(2{\mathbf{p}^{\rm B}},2\tilde{\mathbf{p}}^{\rm
B}|\hat{h^{\pm}})=\sum_{\mu,\nu\in\mathop{\mathrm{missing}}{DP}\nolimits}\hat{h}_{\mu,\nu}^{\pm}Q_{\mu}({\mathbf{p}^{\rm
B}})Q_{\mu}(\tilde{\mathbf{p}}^{\rm B})$ (2.146) $K_{\mu}({\mathbf{p}^{\rm
B}}|\hat{h}^{\pm})=\sum_{\nu\in\mathop{\mathrm{missing}}{DP}\nolimits}\hat{h}_{\mu,\nu}^{\pm}Q_{\nu}({\mathbf{p}^{\rm
B}}))$ (2.147)

which results from (2.115). For a given $\hat{h}^{\pm}$ we obtain

$\tau(2{\mathbf{p}^{\rm B}},2\tilde{\mathbf{p}}^{\rm
B}|\hat{h}^{\pm})=\sum_{\mu\in\mathop{\mathrm{missing}}{DP}\nolimits}2^{-\ell(\mu)}K_{\mu}({\mathbf{p}^{\rm
B}}|\hat{h}^{\pm})Q_{\nu}(\tilde{\mathbf{p}}^{\rm B}))$ (2.148)

For $\hat{h}^{\pm}=1$ (the same: $B=0$) we get the simplest two-sided BKP tau
function (2.113)-(2.114).

######  Remark 2.16.

In the Grassmannian approach to the BKP tau functions developed in [7] the
coefficients $K_{\lambda}({\mathbf{p}^{\rm B}}|\hat{h}^{\pm})$ in (2.142) have
the meaning of the Cartan coordinates $\kappa_{\lambda}(\hat{h}^{\pm})$. The
notation was used in [10] instead of $K_{\lambda}(\mathbf{p}|\hat{h}^{\pm})$,
see Remark 1.5.

#### The known relation between KP and BKP two-sided tau functions.

It is well-known [6] that the square of a (one-side) BKP tau function is equal
to a certain (one-side) KP tau function under restriction $p_{2m}\equiv 0$;
the same is obviously true for two-sided tau functions:

######  Theorem 2.4.

$\tau^{\rm B}(2{\mathbf{p}^{\rm B}},2\tilde{\mathbf{p}}^{\rm
B}|\hat{h}^{+})\tau^{\rm B}(2{\mathbf{p}^{\rm B}},2\tilde{\mathbf{p}}^{\rm
B}|\hat{h}^{-})=\tau(\mathbf{p}^{\prime},\tilde{\mathbf{p}}^{\prime}|\hat{h}^{+}\hat{h}^{-})$
(2.149)

where $\tau^{\rm B}(2{\mathbf{p}^{\rm B}},2\tilde{\mathbf{p}}^{\rm
B}|\hat{h}^{+})=\tau^{\rm B}(2{\mathbf{p}^{\rm B}},2\tilde{\mathbf{p}}^{\rm
B}|\hat{h}^{-})$ and where ${\mathbf{p}^{\rm B}}$ is given in (1.2) and

$\tilde{\mathbf{p}}^{\rm B}=(\tilde{p}^{\rm B}_{1},\tilde{p}^{\rm
B}_{3},\tilde{p}^{\rm B}_{5},\dots)$ (2.150)

(with $2\tilde{p}^{\rm B}_{i}=\tilde{p}_{i},\,i\,{\rm odd}$) are two given
sets of parameters and where

$\mathbf{p}^{\prime}:=(p_{1},0,p_{3},0,p_{5},0,\dots),$ (2.151)
$\tilde{\mathbf{p}}^{\prime}:=(\tilde{p}_{1},0,\tilde{p}_{3},0,\tilde{p}_{5},0,\dots).$
(2.152)

The proof is identical to the similar statement [6] about KP and BKP tau
functions with one set of variables and based on Lemma 2.2 and on the relation

$\hat{\gamma}(\mathbf{p}^{\prime})=\hat{\gamma}^{{\rm B}+}(2{\mathbf{p}^{\rm
B}})\hat{\gamma}^{{\rm B}-}(2{\mathbf{p}^{\rm
B}}),\quad\hat{\gamma}^{\dagger}(\mathbf{p}^{\prime})=\hat{\gamma}^{{\dagger}{\rm
B+}}(2{\mathbf{p}^{\rm B}})\hat{\gamma}^{{\dagger}{\rm B-}}(2{\mathbf{p}^{\rm
B}})$ (2.153)

(which is the direct result of (2.41),(2.44) and can be treated as an example
of application Lemma 2.1).

Our goal will be to get the generalization of (2.149) in Section 3 where now
$\mathbf{p}=\mathbf{p}(\mathbf{a}/-\mathbf{b})$ and
$\tilde{\mathbf{p}}=\tilde{\mathbf{p}}(\tilde{\mathbf{a}}/-\tilde{\mathbf{b}})$
of (2.91 ),(2.93). This choice of power sums is different from (2.151)-(2.152)
except the case mentioned in Remark 2.6.

#### The known relation between KP and BKP lattice tau functions.

In the previous work [10] we have shown that in case the power sums are
restricted according to (2.151) we get

######  Theorem 2.5.

[10]. For ${\mathbf{p}^{\rm B}}$ and $\mathbf{p}$ given respectively by (1.2),
(2.151) and (1.5) we have

$S_{(\alpha|\beta)}(\mathbf{p}|\hat{h}^{+}\hat{h}^{-})=\sum_{\mu\in{\mathcal{P}}(\alpha,\beta)}\,\left[\zeta^{+},\zeta^{-}\atop\alpha,\beta\right]\,K_{\zeta^{+}}(2{\mathbf{p}^{\rm
B}}|\hat{h}^{+})K_{\zeta^{-}}(2{\mathbf{p}^{\rm B}}|\hat{h}^{-})$ (2.154)

where

$\left[\zeta^{+},\zeta^{-}\atop\alpha,\beta\right]:={(-1)^{\tfrac{1}{2}r(r+1)+s}\over
2^{r-s}}\operatorname{sgn}(\nu)(-1)^{\pi(\nu^{-})+\frac{1}{2}m(\hat{\nu}^{-})}:={a}^{\zeta^{+},\zeta^{-}}_{\alpha,-\beta-1}.$
(2.155)

In particular

$s_{(\alpha|\beta)}(\mathbf{p}^{\prime})=\sum_{\mu\in{\mathcal{P}}(\alpha,\beta)}\left[\zeta^{+},\zeta^{-}\atop\alpha,\beta\right]\,Q_{\zeta^{+}}({\mathbf{p}^{\rm
B}})Q_{\zeta^{-}}({\mathbf{p}^{\rm B}})$ (2.156)

obtained in [9].

Proof follows from Lemma 3.6 and Lemma 2.2.

In [10] we consider KP polynomial tau functions as the bilinear sum of BKP
polynomial tau functions [16]; this relation is given by Theorem 2.5.

In Section 4.1, we are going to replace the restriction
$\mathbf{p}=\mathbf{p}^{\prime}$ by
$\mathbf{p}=\mathbf{p}(\mathbf{a}/-\mathbf{b})$ and present the generalization
of Theorem 2.5.

###  2.7 On lattice polynomial KP and lattice polynomial BKP tau functions

Among lattice tau functions there is a family of tau functions polynomial in
power sum variables. Polynomial tau functions were studied in [16] and also in
[11].

To be presize let us introduce

######  Definition 2.7.

We call a lattice KP tau function (2.141) polynomial if for any given
$\lambda\in\mathop{\mathrm{missing}}{P}\nolimits$ the number of matrix
elements $\hat{g}_{\mu,\lambda}$ is finite.

######  Definition 2.8.

We call a lattice BKP tau function (2.147) polynomial if for any given
$\alpha\in\mathop{\mathrm{missing}}{DP}\nolimits$ the number of matrix
elements $\hat{h}_{\nu,\alpha}$ is finite.

######  Remark 2.17.

In the Grassmannain approach the property of polynomiality may be formulated
as follows:

It is known that for any $\lambda\in\mathop{\mathrm{missing}}{P}\nolimits$
there exits such $\hat{g}(\lambda)\in GL_{\infty}$ that
$\hat{g}(\lambda)|0\rangle=\Psi_{\lambda}|0\rangle$ (see Remark 2.12), and
perhaps it will be better to attach a lattice KP tau function (2.139) to the
Schubert cell labelled with the partition $\lambda$. Then one can write

$\hat{g}=\hat{g}_{L}\hat{g}_{0}\hat{g}_{R}\hat{g}(\lambda)$ (2.157)

where $\hat{g}_{L},\hat{g}_{0},\hat{g}_{R}\in GL_{\infty}$ and where

$\hat{g}_{R}|0\rangle=|0\rangle,\quad\langle 0|\hat{g}_{L}=\langle
0|,\quad\hat{g}_{0}|0\rangle=g_{0}|0\rangle,\quad\langle
0|\hat{g}_{0}=g_{0}\langle 0|$ (2.158)

where $g_{0}$ is a number. Then the condition of the polynomiality is the
nilpotent structure of $\hat{A}_{L}$ where $\hat{g}_{L}=e^{\hat{A}_{L}}$.

The similar condition of the polynomiality for (2.145) we get for
$\hat{h}^{\pm}=\hat{h}^{\pm}_{L}\hat{h}^{\pm}_{0}\hat{h}^{\pm}_{R}\hat{h}^{\pm}(\alpha)$
where $\hat{h}^{\pm}(\alpha)|0\rangle=\Phi^{\pm}_{\hat{\alpha}}|0\rangle$ (see
[7]) and where

$\hat{h}^{\pm}_{R}|0\rangle=|0\rangle,\quad\langle 0|\hat{h}^{\pm}_{L}=\langle
0|,\quad\hat{h}^{\pm}_{0}|0\rangle=h_{0}|0\rangle,\quad\langle
0|\hat{h}^{\pm}_{0}=h_{0}\langle 0|$ (2.159)

where $h_{0}$ is a number. Then the condition of the polynomiality is the
nilpotent structure of $\hat{B}^{\pm}_{L}$ where
$\hat{h}^{\pm}_{L}=e^{\hat{B}^{\pm}_{L}}$.

Let us note that in [11] we actually consider only the cases where
$\hat{g}_{L}=1$ and $\hat{h}^{\pm}_{L}=1$.

##  3 Results

###  3.1 Key Lemmas

Let us note that tau functions below depend on
$\tilde{\Psi}(\mathbf{a}/-\mathbf{b})$ and $\tilde{\Phi}(\mathbf{a})$. And as
one can notice, $\tilde{\Psi}(\mathbf{a}/-\mathbf{b})$ and
$\tilde{\Phi}(\mathbf{a})$ do not depend on the order of the arguments
$a_{1},\dots,a_{N},b_{1},\dots,b_{N}$. Nevertheless as we have already wrote
it is suitable to order these sets to have a unified approach for problems
under consideration.

#### Polarization ${\mathcal{P}}(\mathbf{a},\mathbf{b})$ of the coordinate
set.

######  Definition 3.1.

A polarization of the pair of the ordered sets
$(\mathbf{a},\mathbf{b})=(a_{1},\dots,a_{N},b_{1},\dots,b_{N})$, is a pair of
ordered sets $(\mathbf{z}^{+},\mathbf{z}^{-})$ with cardinalities (or lengths)

$N(\mathbf{z}^{+}):=\\#(\mathbf{z}^{+}),\quad
N(\mathbf{z}^{-}):=\\#(\mathbf{z}^{-}),$ (3.1)

satisfying

$\mathbf{z}^{+}\cap\mathbf{z}^{-}=\mathbf{a}\cap\mathbf{b},\quad\mathbf{z}^{+}\cup\mathbf{z}^{-}=\mathbf{a}\cup\mathbf{b}.$
(3.2)

The set of all polarizations of $\mathbf{a},\mathbf{b}$ is denoted
${\mathcal{P}}(\mathbf{a},\mathbf{b})$.

Introduce the following notations:

$\tilde{Q}:=\mathbf{a}\cap\mathbf{b},\quad\tilde{C}:=\mathbf{a}\cup\mathbf{b}$
(3.3)

and their cardinalities as

$\tilde{q}:=\\#(\tilde{Q}),\quad\tilde{c}:=\\#(\tilde{C})=2N-\tilde{q}.$ (3.4)

We have

$N(\mathbf{z}^{+})+N(\mathbf{z}^{-})=2N,$

so $N(\mathbf{z}^{\pm})$ must have the same parity. It is easy to see that the
cardinality of ${\mathcal{P}}(\mathbf{a},\mathbf{b})$ is $2^{2N-2\tilde{q}}$.

######  Lemma 3.1 (Binary sequence associated to a polarization).

For every polarization $\mathbf{z}:=(\mathbf{z}^{+},\mathbf{z}^{-})$ of
$(\mathbf{a},\mathbf{b})$ there is a unique binary sequence of length $2N$

$\epsilon(\mathbf{z})=(\epsilon_{1}(\mathbf{z}),\dots,\epsilon_{2N}(\mathbf{z})),$
(3.5)

with

$\epsilon_{j}(\mathbf{z})=\pm,\quad j=1,\dots 2N,$ (3.6)

such that the sequence of pairs

$((a_{1},\epsilon_{1}(\mathbf{z})),\cdots(a_{N},\epsilon_{N}(\mathbf{z})),(b_{1},\epsilon_{N+1}(\mathbf{z})),\dots,(b_{N},\epsilon_{2N}(\mathbf{z}))$
(3.7)

is a permutation of the sequence

$((z^{+}_{1},+),\cdots(z^{+}_{N(\mathbf{z}^{+})},+),(z^{-}_{1},-),\dots,(z^{-}_{N(\mathbf{z}^{-})},-)).$
(3.8)

######  Definition 3.2.

The sign of the polarization $(\mathbf{z}^{+},\mathbf{z}^{-})$, denoted
$\operatorname{sgn}(\mathbf{z})$, is defined as the sign of the permutation
that takes the sequence (3.7) into the sequence (3.8).

Denote by

$\pi(\mathbf{z}^{\pm}):=\\#(\mathbf{a}\cap\mathbf{z}^{\pm})$ (3.9)

the cardinality of the intersection of $\mathbf{a}$ with $\mathbf{z}^{\pm}$.
It follows that

$\pi(\mathbf{z}^{+})+\pi(\mathbf{z}^{-})=N+\tilde{q}.$ (3.10)

######  Lemma 3.2.

$\Psi(\mathbf{a}/-\mathbf{b})=\sum_{(\mathbf{z}^{+},\mathbf{z}^{-})\in{\mathcal{P}}(\mathbf{a},\mathbf{b})}\textsc{d}^{\mathbf{z}^{+},\mathbf{z}^{-}}_{\mathbf{a},\mathbf{b}}\Phi^{+}(\mathbf{z}^{+})\Phi^{-}(\mathbf{z}^{-}),$
(3.11)
$\tilde{\Psi}(\mathbf{a}/-\mathbf{b})=\sum_{(\mathbf{z}^{+},\mathbf{z}^{-})\in{\mathcal{P}}(\mathbf{a},\mathbf{b})}\tilde{\textsc{d}}^{\mathbf{z}^{+},\mathbf{z}^{-}}_{\mathbf{a},\mathbf{b}}\tilde{\Phi}^{+}(\mathbf{z}^{+})\tilde{\Phi}^{-}(\mathbf{z}^{-}),$
(3.12)

where

$\textsc{d}^{\mathbf{z}^{+},\mathbf{z}^{-}}_{\mathbf{a},\mathbf{b}}:={(-1)^{\tfrac{1}{2}N(N+1)+\tilde{q}}\over
2^{N-\tilde{q}}}\operatorname{sgn}(\mathbf{z})(-1)^{\pi(\mathbf{z}^{-})}i^{N(\mathbf{z}^{-})}$
(3.13)

and

$\tilde{\textsc{d}}^{\mathbf{z}^{+},\mathbf{z}^{-}}_{\mathbf{a},\mathbf{b}}=\textsc{d}^{\mathbf{z}^{+},\mathbf{z}^{-}}_{\mathbf{a},\mathbf{b}}\frac{\Delta^{\rm
B}(\mathbf{z}^{+})\Delta^{\rm
B}(\mathbf{z}^{-})}{\Delta(\mathbf{a},\mathbf{b})}.$ (3.14)

###### Proof.

Consider eq. (2.70) and substitute

$\psi(a_{j})=\tfrac{1}{\sqrt{2}}(\phi^{+}(a_{j})-i\phi^{-}(a_{j})),\quad\psi^{\dagger}(-b_{j})=\tfrac{1}{b_{j}\sqrt{2}}(\phi^{+}(b_{j})+i\phi^{-}(b_{j})),\quad
j=1,\dots,N,$ (3.15)

see (2.22), for all the factors, and expand the product as a sum over monomial
terms of the form

$\prod_{j=1}^{N(\mathbf{z}^{+})}\phi^{+}(z^{+}_{j})\prod_{k=1}^{N(\mathbf{z}^{-})}\phi^{-}(z^{-}_{k}).$
(3.16)

We have in mind formula (2.26) if $b_{k}=a_{j}$ which reduces the number of
terms in the product of Fermi fileds.

Taking into account the sign factor $\operatorname{sgn}(\mathbf{z})$
corresponding to the order of the neutral fermion factors, as well as the
powers of $-1$ and $i$, and noting that there are $2^{\tilde{q}}$ resulting
identical terms, then gives (3.11). Then (3.12) ∎

#### Polarization ${\mathcal{P}}(\mathbf{b}^{*},\mathbf{a}^{*})$.

To begin with let me remind notations (2.51)-(LABEL:bold-x*y*) and also
equalities (2.60)-(2.60=) and (2.62). We repeat the previous paragraph
replacing the set $(\mathbf{a},\mathbf{b})$ by the set
$(\mathbf{b}^{*},\mathbf{a}^{*})$.

######  Definition 3.3.

A polarization of the pair of the ordered sets

$(\mathbf{b}^{*},\mathbf{a}^{*})=(b^{*}_{1},\dots,b^{*}_{N},a^{*}_{1},\dots,a^{*}_{N}),$

is a pair of ordered sets $(\mathbf{z}^{+},\mathbf{z}^{-})$ with cardinalities
(or lengths)

$N(\mathbf{z}^{+}):=\\#(\mathbf{z}^{+}),\quad
N(\mathbf{z}^{-}):=\\#(\mathbf{z}^{-}),$ (3.17)

satisfying

$\mathbf{z}^{+}\cap\mathbf{z}^{-}=\mathbf{b}^{*}\cap\mathbf{a}^{*},\quad\mathbf{z}^{+}\cup\mathbf{z}^{-}=\mathbf{b}^{*}\cup\mathbf{a}^{*}.$
(3.18)

The set of all polarizations of $\mathbf{b}^{*},\mathbf{a}^{*}$ is denoted
${\mathcal{P}}(\mathbf{b}^{*},\mathbf{a}^{*})$.

Denote two sets

$Q:=\mathbf{b}^{*}\cap\mathbf{a}^{*},\quad
C:=\mathbf{b}^{*}\cup\mathbf{a}^{*}$ (3.19)

and its cardinalities as

$q:=\\#(Q),\quad c:=\\#(C)=2N-q.$ (3.20)

We have

$N(\mathbf{z}^{+})+N(\mathbf{z}^{-})=2N,$

so $N(\mathbf{z}^{\pm})$ must have the same parity. It is easy to see that the
cardinality of ${\mathcal{P}}(\mathbf{b}^{*},\mathbf{a}^{*})$ is $2^{2N-2q}$.

######  Lemma 3.3 (Binary sequence associated to a polarization).

For every polarization $\mathbf{z}:=(\mathbf{z}^{+},\mathbf{z}^{-})$ of
$(\mathbf{b}^{*},\mathbf{a}^{*})$ there is a unique binary sequence of length
$2N$

$\epsilon(\mathbf{z})=(\epsilon_{1}(\mathbf{z}),\dots,\epsilon_{2N}(\mathbf{z}))$
(3.21)

with

$\epsilon_{j}(\mathbf{z})=\pm,\quad j=1,\dots 2N,$ (3.22)

such that the sequence of pairs

$((b^{*}_{1},\epsilon_{1}(\mathbf{z})),\cdots(b^{*}_{N},\epsilon_{N}(\mathbf{z})),(a^{*}_{1},\epsilon_{N+1}(\mathbf{z})),\dots,(a^{*}_{N},\epsilon_{2N}(\mathbf{z}))$
(3.23)

is a permutation of the sequence

$((z^{+}_{1},+),\cdots(z^{+}_{N(\mathbf{z}^{+})},+),(z^{-}_{1},-),\dots,(z^{-}_{N(\mathbf{z}^{-})},-)).$
(3.24)

######  Definition 3.4.

The sign of the polarization $(\mathbf{z}^{+},\mathbf{z}^{-})$, denoted
$\operatorname{sgn}(\mathbf{z})$, is defined as the sign of the permutation
that takes the sequence (3.23) into the sequence (3.24).

Denote by

$\pi(\mathbf{z}^{\pm}):=\\#(\mathbf{b}^{*}\cap\mathbf{z}^{\pm})$ (3.25)

the cardinality of the intersection of $\mathbf{b}^{*}$ with
$\mathbf{z}^{\pm}$. It follows that

$\pi(\mathbf{z}^{+})+\pi(\mathbf{z}^{-})=N+q.$ (3.26)

######  Lemma 3.4.

$\Psi(\mathbf{b}^{*}/-\mathbf{a}^{*})=\sum_{(\mathbf{z}^{+},\mathbf{z}^{-})\in{\mathcal{P}}(\mathbf{b}^{*},\mathbf{a}^{*})}\textsc{d}^{\mathbf{z}^{+},\mathbf{z}^{-}}_{\mathbf{b}^{*},\mathbf{a}^{*}}\Phi^{+}(\mathbf{z}^{+})\Phi^{-}(\mathbf{z}^{-}),$
(3.27)
$\tilde{\Psi}(\mathbf{b}^{*}/-\mathbf{a}^{*})=\sum_{(\mathbf{z}^{+},\mathbf{z}^{-})\in{\mathcal{P}}(\mathbf{b}^{*},\mathbf{a}^{*})}\tilde{\textsc{d}}^{\mathbf{z}^{+},\mathbf{z}^{-}}_{\mathbf{b}^{*},\mathbf{a}^{*}}\tilde{\Phi}^{+}(\mathbf{z}^{+})\tilde{\Phi}^{-}(\mathbf{z}^{-}),$
(3.28)

where

$\textsc{d}_{\mathbf{b}^{*},\mathbf{a}^{*}}^{\mathbf{z}^{+},\mathbf{z}^{-}}:={(-1)^{\tfrac{1}{2}N(N+1)+q}\over
2^{N-q}}\operatorname{sgn}(\mathbf{z})(-1)^{\pi(\mathbf{z}^{+})}i^{N(\mathbf{z}^{+})}$
(3.29)

and

$\tilde{\textsc{d}}^{\mathbf{z}^{+},\mathbf{z}^{-}}_{\mathbf{b}^{*},\mathbf{a}^{*}}=\textsc{d}^{\mathbf{z}^{+},\mathbf{z}^{-}}_{\mathbf{b}^{*},\mathbf{a}^{*}}\frac{\Delta^{\rm
B}(\mathbf{z}^{+})\Delta^{\rm
B}(\mathbf{z}^{-})}{\Delta(\mathbf{b}^{*}/-\mathbf{a}^{*})}.$ (3.30)

###### Proof.

Consider eq. (2.70) and substitute

$\psi(-a_{j}^{-1})=\tfrac{1}{\sqrt{2}}(\phi^{+}(-a_{j}^{-1})-i\phi^{-}(-a_{j}^{-1})),\quad\psi^{\dagger}(b_{j}^{-1})=\tfrac{b_{j}}{\sqrt{2}}(\phi^{+}(-b_{j}^{-1})+i\phi^{-}(-b_{j}^{-1})),\quad
j=1,\dots,N,$ (3.31)

see (2.22), for all the factors, and expand the product as a sum over monomial
terms of the form

$\prod_{j=1}^{N(\mathbf{z}^{+})}\phi^{+}(z^{+}_{j})\prod_{k=1}^{N(\mathbf{z}^{-})}\phi^{-}(z^{-}_{k}).$
(3.32)

We have in mind formula (2.27) if ${b}_{k}={a}_{j}$ which reduces the number
of terms in the product of Fermi fileds.

Taking into account the sign factor $\operatorname{sgn}(\mathbf{z})$
corresponding to the order of the neutral fermion factors, as well as the
powers of $-1$ and $i$, and noting that there are $2^{q}$ resulting identical
terms, then gives (3.27). Then (3.28) follows. ∎

#### Polarization of ${\mathcal{P}}(\alpha,\beta)$ the set of the Frobenius
coordinates [10].

This paragraph is taken from [10].

######  Definition 3.5.

A polarization of $(\alpha|\beta)$, is a pair $(\zeta^{+},\zeta^{-})$ of
strict partitions with cardinalities (or lengths)

$m(\zeta^{+}):=\\#(\zeta^{+}),\quad m(\zeta^{-})):=\\#(\zeta^{-})$ (3.33)

(including possibly a zero part $\zeta^{+}_{m(\zeta^{+})}=0$ or
$\zeta^{-}_{m(\zeta^{-})}=0$), satisfying

$\zeta^{+}\cap\zeta^{-}=\alpha\cap
I(\beta),\quad\zeta^{+}\cup\zeta^{-}=\alpha\cup I(\beta),$ (3.34)

where

$I(\beta):=(I_{1}(\beta),\dots I_{r}(\beta))$ (3.35)

is the strict partition [17] with parts

$I_{j}(\beta)=\beta_{j}+1,\quad j=1,\dots r.$ (3.36)

The set of all polarizations of $(\alpha|\beta)$ is denoted
${\mathcal{P}}(\alpha,\beta)$.

We denote the strict partition obtained by intersecting $\alpha$ with
$I(\beta)$ as

$S:=\alpha\cap I(\beta)$ (3.37)

and its cardinality as

$s:=\\#(S).$ (3.38)

Since both $\alpha$ and $I(\beta)$ have cardinality $r$, it follows that

$m(\zeta^{+})+m(\zeta^{-})=2r,$ (3.39)

so $m(\zeta^{\pm})$ must have the same parity. It is easily verified [9] that
the cardinality of ${\mathcal{P}}(\alpha,\beta)$ is $2^{2r-2s}$. The following
was proved in [9].

######  Lemma 3.5 (Binary sequence associated to a polarization).

For every polarization $\mu:=(\zeta^{+},\zeta^{-})$ of
$\lambda=(\alpha|\beta)$ there is a unique binary sequence of length $2r$

$\epsilon(\mu)=(\epsilon_{1}(\mu),\dots,\epsilon_{2r}(\mu)),$ (3.40)

with

$\epsilon_{j}(\mu)=\pm,\quad j=1,\dots 2r,$ (3.41)

such that

  1. 1.

The sequence of pairs

$((\alpha_{1},\epsilon_{1}(\mu)),\cdots(\alpha_{r},\epsilon_{r}(\mu)),(\beta_{1}+1,\epsilon_{r+1}(\mu)),\dots,(\beta_{r}+1,\epsilon_{2r}(\mu))$
(3.42)

is a permutation of the sequence

$((\zeta^{+}_{1},+),\cdots(\zeta^{+}_{m^{+}(\mu)},+),(\zeta^{-}_{1},-),\dots,(\zeta^{-}_{m^{-}(\mu)},-)$
(3.43)

  2. 2.

$\epsilon_{j}(\mu)=+\ \text{ if }\alpha_{j}\in S,\ \text{ and
}\epsilon_{r+j}(\mu)=-\ \text{ if }\beta_{j}+1\in S,\quad j=1,\dots,r.$ (3.44)

######  Definition 3.6.

The sign of the polarization $(\zeta^{+},\zeta^{-})$, denoted
$\operatorname{sgn}(\mu)$, is defined as the sign of the permutation that
takes the sequence (3.42) into the sequence (3.43).

Denote by

$\pi(\zeta^{\pm}):=\\#(\alpha\cup\zeta^{\pm})$ (3.45)

the cardinality of the intersection of $\alpha$ with $\zeta^{\pm}$. It follows
that

$\pi(\zeta^{+})+\pi(\zeta^{-})=r+s.$ (3.46)

For $\lambda=(\alpha|\beta)$, we have

######  Lemma 3.6.

$\Psi_{\lambda}={(-1)^{\tfrac{1}{2}r(r+1)+s}\over
2^{r-s}}\sum_{(\zeta^{+},\zeta^{-})\in{\mathcal{P}}(\alpha,\beta)}\operatorname{sgn}(\mu)(-1)^{\pi(\zeta^{-})}i^{m(\zeta^{-})}\Phi_{\zeta^{+}}\Phi_{\zeta^{-}}.$
(3.47)

###### Proof.

In eq. (2.64), reorder the product over the factors
$\psi_{\alpha_{j}}\psi^{\dagger}_{-\beta_{j}-1}$ so the $\psi_{\alpha_{j}}$
terms precede the $\psi^{\dagger}_{-\beta_{j}-1}$ ones, giving an overall sign
factor $(-1)^{\tfrac{1}{2}r(r-1)}$. Then substitute

$\psi_{\alpha_{j}}=\tfrac{1}{\sqrt{2}}(\phi^{+}_{\alpha_{j}}-i\phi^{-}_{\alpha_{j}}),\quad\psi^{\dagger}_{-\beta_{j}-1}=\tfrac{(-1)^{j}}{\sqrt{2}}(\phi^{+}_{\beta_{j}+1}+i\phi^{-}_{\beta_{j}+1}),\quad
j\in\mathbf{Z},$ (3.48)

which follows from (2.7), for all the factors, and expand the product as a sum
over monomial terms of the form

$\prod_{j=1}^{m(\zeta^{+})}\phi^{+}_{\zeta^{+}_{j}}\prod_{k=1}^{m(\zeta^{-})}\phi^{-}_{\zeta^{-}_{k}}.$
(3.49)

Taking into account the sign factor $\operatorname{sgn}(\mu)$ 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
(3.47). ∎

For $\lambda=(\alpha|\beta)$, we also have

######  Lemma 3.7.

$\Psi^{\dagger}_{\lambda}={(-1)^{\tfrac{1}{2}r(r+1)+s}\over
2^{r-s}}(-1)^{|\lambda|}\sum_{(\zeta^{+},\zeta^{-})\in{\mathcal{P}}(\alpha,\beta)}\operatorname{sgn}(\mu)(-1)^{\pi(\zeta^{-})}i^{m(\zeta^{-})}\Phi_{-\zeta^{+}}\Phi_{-\zeta^{-}}.$
(3.50)

##  4 Bilinear expressions

Below we imply the condition

$\hat{g}=\hat{h}^{+}\hat{h}^{-}$ (4.1)

Below we use Lemma 2.2 and results of Section 3.1.

###  4.1  KP tau functions as a bilinear expression in BKP tau functions

Let

$\displaystyle
p_{m}(\mathbf{a}/-\mathbf{b})=\sum_{j=1}^{N}\left(a_{j}^{m}-(-b_{j})^{m}\right),\quad
m>0,$ (4.2)
$\displaystyle\tilde{p}_{m}(\tilde{\mathbf{a}}/-\tilde{\mathbf{b}})=\sum_{j=1}^{\tilde{N}}\left(\tilde{a}_{j}^{m}-(-\tilde{b}_{j})^{m}\right),\quad
m>0.$ (4.3)

For a given $\mathbf{z}=(z_{1},\dots,z_{N(\mathbf{z})})$ we define

$\displaystyle p^{\rm
B}_{N}(\mathbf{z})=\sum_{j=1}^{N(\mathbf{z})}z_{j}^{m},\quad m>0,$ (4.4)
$\displaystyle\tilde{p}^{\rm
B}_{m}(\tilde{\mathbf{z}})=\sum_{j=1}^{m(\tilde{\mathbf{z}})}\tilde{z}_{j}^{m},\quad
m>0.$ (4.5)

#### Two-sided KP tau function as a bilinear expression in two-sided BKP tau
functions.

Then

$\tau(\mathbf{p}(\mathbf{a}/-\mathbf{b}),\tilde{\mathbf{p}}(\tilde{\mathbf{a}}/-\tilde{\mathbf{b}})|\hat{g})=\langle
0|\,\tilde{\Psi}(\mathbf{b}^{*}/-\mathbf{a}^{*})\,\hat{g}\,\tilde{\Psi}(\tilde{\mathbf{a}}/-\tilde{\mathbf{b}})|0\rangle=:\tau(\mathbf{a}/-\mathbf{b};\tilde{\mathbf{a}}/-\tilde{\mathbf{b}}|\hat{g})$
(4.6)

and

$\tau^{{\rm B}\pm}({\mathbf{p}^{\rm B}}(\mathbf{z}),\tilde{\mathbf{p}}^{\rm
B}(\tilde{\mathbf{z}})|\hat{h}^{\pm}):=\langle
0|\,\tilde{\Phi}^{\pm}(-\mathbf{z}^{*})\hat{h}^{\pm}\tilde{\Phi}^{\pm}(\tilde{\mathbf{z}})|0\rangle=:\tau^{{\rm
B}\pm}(\mathbf{z}|\tilde{\mathbf{z}}|\hat{h}^{\pm}).$ (4.7)

######  Theorem 4.1.

Suppose $\hat{g}=\hat{h}^{+}\hat{h}^{-}$. Then

$\tau(\mathbf{a}/-\mathbf{b};\tilde{\mathbf{a}}/-\tilde{\mathbf{b}}|\hat{h}^{+}\hat{h}^{-})=\sum_{{\mathbf{z}}\in{\mathcal{P}}({\mathbf{a}},{\mathbf{b}})\atop\tilde{\mathbf{z}}\in{\mathcal{P}}(\tilde{\mathbf{a}},\tilde{\mathbf{b}})}\left[\mathbf{z}^{-},\mathbf{z}^{+}\atop\mathbf{a},\mathbf{b}\right]\left[\tilde{\mathbf{z}}^{-},\tilde{\mathbf{z}}^{+}\atop\tilde{\mathbf{a}},\tilde{\mathbf{b}}\right]\,\tau^{{\rm
B}\pm}(\mathbf{z}^{+};\tilde{\mathbf{z}}^{+}|\hat{h}^{+})\tau^{{\rm
B}\pm}(\mathbf{z}^{-};\tilde{\mathbf{z}}^{-}|\hat{h}^{-})$ (4.8)

where

$\displaystyle\left[\mathbf{z}^{+},\mathbf{z}^{-}\atop\mathbf{a},\mathbf{b}\right]=\frac{\Delta^{\rm
B}(\mathbf{z}^{+})\Delta^{\rm
B}(\mathbf{z}^{-})}{\Delta(\mathbf{b}^{*}/-\mathbf{a}^{*})}{(-1)^{\tfrac{1}{2}N(N+1)+q}\over
2^{N-q}}\operatorname{sgn}(\mathbf{z})(-1)^{\pi(\mathbf{z}^{-})+\frac{1}{2}m(\hat{\mathbf{z}}^{-})}\,,$
(4.9)
$\displaystyle\left[\tilde{\mathbf{z}}^{+},\tilde{\mathbf{z}}^{-}\atop\tilde{\mathbf{a}},\tilde{\mathbf{b}}\right]=(-1)^{\tilde{N}^{2}}\frac{\Delta^{\rm
B}(\tilde{\mathbf{z}}^{+})\Delta^{\rm
B}(\tilde{\mathbf{z}}^{-})}{\Delta(\tilde{\mathbf{a}}/-\tilde{\mathbf{b}})}{(-1)^{\tfrac{1}{2}\tilde{N}(\tilde{N}+1)+\tilde{q}}\over
2^{\tilde{N}-\tilde{q}}}\operatorname{sgn}(\tilde{\mathbf{z}})(-1)^{\pi(\tilde{\mathbf{z}}^{-})+\frac{1}{2}m(\hat{\tilde{\mathbf{z}}}^{-})}\prod_{i=1}^{\tilde{N}}\tilde{a}_{i}\tilde{b}_{i}\,.$
(4.10)

Proof follows from Lemma 3.2, Lemma 3.4 and Lemma 2.2.

######  Corollary 4.2.

For $\hat{g}=\hat{h}^{\pm}=1$ we obtain

Example. Let us consider the case $\hat{g}=1$ (that is
$\hat{g}_{\mu,\lambda}=\delta_{\mu,\lambda}$ for
$\mu,\lambda\in\mathop{\mathrm{missing}}{P}\nolimits$) and
$\mathbf{b}=\mathbf{a}$, $\tilde{\mathbf{a}}=\tilde{\mathbf{b}}$, $N=1$.

$\tau(\mathbf{a}/-\mathbf{b}|\tilde{\mathbf{a}}/-\tilde{\mathbf{b}})=\frac{1}{(x+y)(\tilde{x}+\tilde{y})}$
(4.11)

$z^{\pm}_{1}=x$, $z^{\mp}_{2}=y$, $\tilde{z}^{\pm}_{1}=\tilde{x}$,
$\tilde{z}^{\mp}_{2}=\tilde{y}$.

#### Lattice KP tau function as a bilinear expression in the lattice BKP tau
functions.

Denote

$\pi_{(\alpha|\beta)}(\hat{g})(\mathbf{a}/-\mathbf{b}):=\langle
0|\,\tilde{\Psi}(\mathbf{b}^{*}/-\mathbf{a}^{*})\,\hat{g}\,\Psi_{\lambda}\,|0\rangle,$
(4.12)

labelled by partitions $\lambda=(\alpha|\beta)$ and a $B_{\infty}$ lattice of
BKP $\tau$-functions

$\kappa_{\alpha}(\hat{h})(\mathbf{z}):=\langle
0|\,\tilde{\Phi}^{\pm}(-\mathbf{z}^{*})\,\hat{h}^{\pm}\,\Phi^{\pm}_{\alpha}\,|0\big{)}.$
(4.13)

######  Remark 4.1.

The case $\lambda=0$ describes the usual (one-side) tau function.

######  Theorem 4.3.

Suppose $\hat{g}=\hat{h}^{+}\hat{h}^{-}$. Then

$\pi_{(\alpha|\beta)}(g)(\mathbf{a}/-\mathbf{b})=\sum_{\mathbf{z}\in{\mathcal{P}}(\mathbf{b}^{*},\mathbf{a}^{*})\atop\mu\in{\mathcal{P}}(\alpha,\beta)}\,\left[\mathbf{z}^{+},\mathbf{z}^{-}\atop\mathbf{a},\mathbf{b}\right]\left[\zeta^{+},\zeta^{-}\atop\alpha,\beta\right]\kappa_{\zeta^{+}}(h)(\mathbf{z}^{+})\kappa_{\zeta^{-}}(h)(\mathbf{z}^{-})$
(4.14)

where

$\displaystyle\left[\mathbf{z}^{+},\mathbf{z}^{-}\atop\mathbf{a},\mathbf{b}\right]=\frac{\Delta^{\rm
B}(\mathbf{z}^{+})\Delta^{\rm
B}(\mathbf{z}^{-})}{\Delta(\mathbf{b}^{*}/-\mathbf{a}^{*})}{(-1)^{\tfrac{1}{2}N(N+1)}\over
2^{N}}\operatorname{sgn}(\mathbf{z})(-1)^{\pi(\mathbf{z}^{-})+\frac{1}{2}N(\hat{\mathbf{z}}^{-})}\,,$
(4.15)
$\displaystyle\left[\zeta^{+},\zeta^{-}\atop\alpha,\beta\right]={(-1)^{\tfrac{1}{2}r(r+1)+s}\over
2^{r-s}}\operatorname{sgn}(\nu)(-1)^{\pi(\nu^{-})+\frac{1}{2}m(\hat{\nu}^{-})}\,.$
(4.16)

In particular ($\hat{g}=1$)

$s_{(\alpha|\beta)}(\mathbf{a}/-\mathbf{b})=\sum_{\mathbf{z}\in{\mathcal{P}}(\mathbf{b}^{*},\mathbf{a}^{*})\atop\mu\in{\mathcal{P}}(\alpha,\beta)}\,\left[\mathbf{z}^{+},\mathbf{z}^{-}\atop\mathbf{a},\mathbf{b}\right]\left[\zeta^{+},\zeta^{-}\atop\alpha,\beta\right]Q_{\zeta^{+}}(\mathbf{z}^{+})Q_{\zeta^{-}}(\mathbf{z}^{-}).$
(4.17)

Proof follows from Lemma 3.6, Lemma 3.4 and Lemma 2.2.

####  Bi-lattice KP and bi-lattice BKP tau functions.

In view of the notion of the lattice tau functions (1.11) and (1.12) it is
natural to call $\hat{g}_{\lambda,\tilde{\lambda}}$ bi-lattice KP tau function
and to call $\hat{h}_{\mu,\tilde{\mu}}$ bi-lattice BKP tau function. Then we
get

######  Theorem 4.4.

$\hat{g}_{(\alpha|\beta)),(\tilde{\alpha}|\tilde{\beta})}=\sum_{(\zeta^{+},\zeta^{-})\in{\mathcal{P}}(\alpha,\beta)\atop(\tilde{\zeta}^{+},\tilde{\zeta}^{-})\in{\mathcal{P}}(\tilde{\alpha},\tilde{\beta})}\left[\tilde{\zeta}^{+},\tilde{\zeta}^{-}\atop\tilde{\alpha},\tilde{\beta}\right]\left[\zeta^{+},\zeta^{-}\atop\alpha,\beta\right]\hat{h}^{-}_{\zeta^{-},\tilde{\zeta}^{-}}\hat{h}^{+}_{\zeta^{+},\tilde{\zeta}^{+}}$
(4.18)

where

$\displaystyle\left[\tilde{\zeta}^{+},\tilde{\zeta}^{-}\atop\tilde{\alpha},\tilde{\beta}\right]=$
(4.19) $\displaystyle\left[\zeta^{+},\zeta^{-}\atop\alpha,\beta\right]=$
(4.20)

where $\zeta^{-}\cup\zeta^{+}=\alpha\cup(\beta+1)$ and
$\tilde{\zeta}^{-}\cup\tilde{\zeta}^{+}=\tilde{\alpha}\cup(\tilde{\beta}+1)$
and where $\lambda=(\alpha|\beta)$,
$\tilde{\lambda}=(\tilde{\alpha}|\tilde{\beta}))$ and
$\zeta^{\pm}=(\zeta^{\pm}_{1},\dots,\zeta^{\pm}_{m(\zeta^{\pm})})$ and
$\tilde{\zeta}^{\pm}=(\tilde{\zeta}^{\pm}_{1},\dots,\tilde{\zeta}^{\pm}_{m(\tilde{\zeta}^{\pm})})$
where $\alpha=(\alpha_{1},\dots,\alpha_{r})$,
$\beta=(\beta_{1},\dots,\beta_{r})$,
$\tilde{\alpha}=(\tilde{\alpha}_{1},\dots,\tilde{\alpha}_{\tilde{r}})$,
$\tilde{\beta}=(\tilde{\beta}_{1},\dots,\tilde{\beta}_{\tilde{r}})$ and where
$m(\zeta^{-})+m(\zeta^{+})=r$,
$m(\tilde{\zeta}^{-})+m(\tilde{\zeta}^{+})=\tilde{r}$. The weights denoted by
square brackets will be written below.

######  Corollary 4.5.

For $\hat{g}=1$ we have

$\delta_{\alpha,\tilde{\alpha}}\delta_{\beta\tilde{\beta}}=\sum_{(\zeta^{+},\zeta^{-})\in{\mathcal{P}}(\alpha,\beta)\atop(\tilde{\zeta}^{+},\tilde{\zeta}^{-})\in{\mathcal{P}}(\tilde{\alpha},\tilde{\beta})}\left[\tilde{\zeta}^{+},\tilde{\zeta}^{-}\atop\tilde{\alpha},\tilde{\beta}\right]\left[\zeta^{+},\zeta^{-}\atop\alpha,\beta\right]\delta_{\zeta^{-},\tilde{\zeta}^{-}}\delta_{\zeta^{+},\tilde{\zeta}^{+}}$
(4.21)

The simple nontrivial example of the bi-lattice KP tau function
$\hat{g}_{\lambda,\tilde{\lambda}}$ is the product
$s^{*}_{\mu}(\lambda)\dim\,\lambda$ where $s^{*}_{\mu}(\lambda)$ is the so-
called shifted Schur function introduced by Okounkov in [4] and $\dim\lambda$
is the number of standard tablauox of the shape $\lambda$, see [17]. The
shifted Schur functions were used in an approach to the representation theory
developed by G.Olshanski and A.Okounkov in [29].

The simple nontrivial example of the bi-lattice BKP tau function $h_{\mu,\nu}$
is the product $Q^{*}_{\mu}(\nu)\dim^{\rm B}\mu$ where $Q^{*}_{\mu}(\nu)$ is
the shifted projective Schur function introduced by Ivanov in [12] and
$\dim^{\rm B}\mu$ is the number of the shifted standard tablauex of the shape
$\mu$. Functions $Q^{*}_{\mu}(\nu)$ are of use in the study of spin Hurwitz
numbers [18].

The relation between shifted Schur functions and the shifted projective Schur
functions was written done in [30].

##  5 Summary of other formulas [30]

###  5.1 Relation between characters of symmetric group and characters of
Sergeev group

For a given $\Delta\in\mathop{\mathrm{missing}}{OP}\nolimits$, one can split
its parts into two ordered odd partitions $(\Delta^{+},\Delta^{-})$:
$\Delta=\Delta^{+}\cup\Delta^{-}$, $\Delta^{+},\Delta^{-}\in OP$,
$\ell(\Delta^{+})+\ell(\Delta^{-})=\ell(\Delta)$. The set of all such
$(\Delta^{+},\Delta^{-})$ we denote $O{\mathcal{P}}(\Delta)$.

From $J_{n}=J^{\rm B+}_{n}+J^{\rm B^{-}}_{n},\,n\,{\rm odd}$ (see [6]), we
obtain

######  Lemma 5.1.

$J_{\Delta}=\sum_{(\Delta^{+},\Delta^{-})\in O{\mathcal{P}}}J^{{\rm
B}+}_{\Delta^{+}}J^{{\rm B}-}_{\Delta^{-}},\qquad
J_{-\Delta}=\sum_{\Delta^{+}\in\mathop{\mathrm{missing}}{OP}\nolimits\atop\Delta^{+}\cup\Delta^{+}=\Delta}J^{{\rm
B}+}_{-\Delta^{+}}J^{{\rm B}-}_{-\Delta^{-}},$ (5.1)

It is known [17] that the power sums labeled by partitions:

$\mathbf{p}_{\Delta}=p_{\Delta_{1}}p_{\Delta_{2}}\cdots,\quad\Delta\in\mathop{\mathrm{missing}}{P}\nolimits$
(5.2)

(here $\mathbf{p}=(p_{1},p_{2},p_{3},\dots)$) are uniquely expressed in terms
of Schur polynomials

$\mathbf{p}_{\Delta}=\sum_{\lambda\in\mathop{\mathrm{missing}}{P}\nolimits}\chi_{\lambda}(\Delta)s_{\lambda}(\mathbf{p}),\quad$
(5.3)

while the odd power sum variables (power sums labels by odd numbers)

$\mathbf{p}_{\Delta}=p_{\Delta_{1}}p_{\Delta_{2}}\cdots,\quad\Delta\in\mathop{\mathrm{missing}}{OP}\nolimits$
(5.4)

also denoted $\mathbf{p}^{\rm B}_{\Delta}$ (where ${\mathbf{p}^{\rm
B}}=(p_{1},p_{3},p_{5},\dots)$) are uniquely expressed in terms of projective
Schur polynomials:

$\mathbf{p}^{\rm
B}_{\Delta}=\sum_{\alpha\in\mathop{\mathrm{missing}}{DP}\nolimits}\chi^{\rm
B}_{\alpha}(\Delta)Q_{\alpha}({\mathbf{p}^{\rm
B}})=\mathbf{p}_{\Delta},\quad\Delta\in\mathop{\mathrm{missing}}{OP}\nolimits$
(5.5)

Let me recall that the coefficients $\chi_{\lambda}(\Delta)$ in (5.3) has the
meaning of the irreducible characters of the symmetric group
$S_{d},\,d=|\lambda|$ evaluated on the cycle class $C_{\Delta}$
$\Delta=(\Delta_{1},\dots,\Delta_{k})$, $|\lambda|=|\Delta|=d$, see [17], and
we can write it as

$\chi_{\lambda}(\Delta)=\langle 0|J_{\Delta}\Psi_{\lambda}|0\rangle$ (5.6)

where $J_{\Delta}$ and $\Psi_{\lambda}$ are given respectively by (2.87) and
(2.63) (see for instance [20] for details).

The characters of symmetric groups they have a very wide application, in
particular, in mathematical physics. I will give two works as an example
[5],[22].

The notion of Sergeev group was introduced in [4], see Appendix. The
coefficient $\chi^{\rm B}_{\alpha}$ is the irreducible character of this group
[4],[34]. As it was shown in [4] the so-called spin Hurwitz numbers
(introduced in this work) are expressed in terms of these characters. As it
was pointed out in [13],[18] the generating function for spin Hurwitz numbers
can be related to the BKP hierarchy in a similar way as the generating
function for usual Hurwitz numbers is related to the KP (and also to Toda
lattice) hierarchies [26],[27], [19].

We can write these characters in terms of the BKP currents $J^{\rm B}_{m}$
($m$ odd):

$\chi^{\rm B}_{\alpha}(\Delta)=2^{-\ell(\alpha)}\langle 0|J^{\rm
B}_{\Delta}\Phi_{\alpha}|0\rangle$ (5.7)

see [20].

From Lemma 3.6 and (5.1) we obtain

######  Theorem 5.2.

The character $\chi_{\lambda},\,\lambda=(\alpha|\beta)$ evaluated on an odd
cycle $\Delta\in\mathop{\mathrm{missing}}{OP}\nolimits$ is the bilinear
function of the Sergeev characters as follows:

$\chi_{(\alpha|\beta)}(\Delta)=\sum_{(\nu^{+},\nu^{-})\in{\mathcal{P}}(\alpha,\beta)\atop(\Delta^{+},\Delta^{-})\in
O{\mathcal{P}}(\Delta)}2^{\ell(\nu^{+})+\ell(\nu^{+})}a^{\nu^{+},\nu^{-}}_{\alpha,-\beta-1}\chi^{\rm
B+}_{\nu^{+}}(\Delta^{+})\chi^{\rm B-}_{\nu^{-}}(\Delta^{-})$ (5.8)

where $a^{\nu^{+},\nu^{-}}_{\alpha,-\beta-1}$ are given by (2.155).

###  5.2 Relation between generalized skew Schur polynomials and generalized
skew projective Schur polynomials

This section may be treated as a remark to [9].

Let us find the relation between the following quantities:

$\displaystyle s_{\lambda/\mu}(\mathbf{p}^{\prime})$ $\displaystyle:=$
$\displaystyle\langle
0|\Psi^{*}_{\mu}\hat{\gamma}(\mathbf{p}^{\prime})\Psi_{\lambda}|0\rangle,$
(5.9) $\displaystyle Q_{\nu/\theta}({\mathbf{p}^{\rm B}})$ $\displaystyle:=$
$\displaystyle\langle 0|\Phi_{-\theta}\hat{\gamma}^{\rm B}(2{\mathbf{p}^{\rm
B}})\Phi_{\nu}|0\rangle$ (5.10)

where the power sum variables are as follows:
$\mathbf{p}^{\prime}=(p_{1},0,p_{3},0,p_{5},0,\dots)$, ${\mathbf{p}^{\rm
B}}=\frac{1}{2}(p_{1},p_{3},p_{5},\dots)$, where $\lambda=(\alpha|\beta)$,
$\mu=(\hat{\gamma}|\delta)$ and where
$\alpha,\beta,\hat{\gamma},\delta,\theta,\nu\in\mathop{\mathrm{missing}}{DP}\nolimits$.

######  Theorem 5.3.

$s_{\lambda/\mu}(\mathbf{p}^{\prime})=\sum_{(\nu^{+},\nu^{-})\in{\mathcal{P}}(\alpha,\beta)\atop(\theta^{+},\theta^{-})\in{\mathcal{P}}(\hat{\gamma},\delta)}(-1)^{|\theta^{+}|+|\theta^{-}|}a^{\nu^{+},\nu^{-}}_{\alpha,-\beta-1}a^{\theta^{+},\theta^{-}}_{\hat{\gamma},-\delta-1}Q_{\nu^{+}/\theta^{+}}({\mathbf{p}^{\rm
B}})Q_{\nu^{-}/\theta^{-}}({\mathbf{p}^{\rm B}})$ (5.11)

For the proof we apply Lemma 3.6 to the first equality (5.9) and consider all
possible parities of $m(\nu^{\pm}),m(\theta^{\pm})$ to apply Lemma 2.2 and
take into account the fermionic expression (5.10).

Theorem 5.3 in a straightforward way can be generalized for the generalized
skew Schur and skew projective Schur polynomials which we define as follows:
Suppose

$g=g(C)=e^{\sum C_{ij}\psi_{i}\psi^{\dagger}_{j}}$ (5.12)

where the entries $C_{ij}$ form a matrix. Also suppose

$h^{\pm}=h^{\pm}(A)=e^{\sum A_{ij}\phi_{i}^{\pm}\phi_{j}^{\pm}}$ (5.13)

where now the entries $A_{ij}=-A_{ji}$ form a skew-symmetric matrix. Suppose
that

$g|0\rangle=|0\rangle c_{1},\quad h^{\pm}|0\rangle=|0\rangle c_{2},\quad
c_{1,2}\in\mathbb{C},\quad c_{1,2}\neq 0.$ (5.14)

Define the generalized Schur polynomials and the generalized projective Schur
polynomials by

$\displaystyle s_{\lambda/\mu}(\mathbf{p}|g)$ $\displaystyle:=$
$\displaystyle\langle\mu|\hat{\gamma}(\mathbf{p})g|\lambda\rangle,$ (5.15)
$\displaystyle Q_{\nu/\theta}({\mathbf{p}^{\rm B}}|h^{\pm})$ $\displaystyle:=$
$\displaystyle\langle 0|\Phi_{-\theta}\hat{\gamma}^{\rm
B\pm}(2{\mathbf{p}^{\rm B}})h^{\pm}\Phi_{\nu}|0\rangle$ (5.16)

######  Remark 5.1.

In ([10]) $s_{\lambda/\mu}(\mathbf{p}|g)$ was defined
$s_{\lambda/\mu}(\mathbf{p}|C)$ and $Q_{\nu/\theta}({\mathbf{p}^{\rm
B}}|h^{\pm})$ was defined $Q_{\nu/\theta}({\mathbf{p}^{\rm B}}|A)$.

The polynomiality of (5.15) in $p_{1},\dots,p_{|\lambda|-|\mu|}$ and the
polynomiality of (5.16) in $p_{1},\dots,p_{|\nu|-|\theta|}$ follows from
(5.14).

One can treat a given $\mathbf{p}$ as a constant and study
$s_{\lambda/\mu}(\mathbf{p}|A)$ as the function of discrete variables
$\lambda_{i}-i$ and $\mu_{i}-i$.

######  Theorem 5.4.

Suppose $g=h^{+}h^{-}$ and (5.14) is true. Then

$s_{\lambda/\mu}(\mathbf{p}^{\prime}|g)=\sum_{(\nu^{+},\nu^{-})\in{\mathcal{P}}(\alpha,\beta)\atop(\theta^{+},\theta^{-})\in{\mathcal{P}}(\hat{\gamma},\delta)}(-1)^{|\theta^{+}|+|\theta^{-}|}a^{\nu^{+},\nu^{-}}_{\alpha,-\beta-1}a^{\theta^{+},\theta^{-}}_{\hat{\gamma},-\delta-1}Q_{\nu^{+}/\theta^{+}}({\mathbf{p}^{\rm
B}}|h^{+})Q_{\nu^{-}/\theta^{-}}({\mathbf{p}^{\rm B}}|h^{-})$ (5.17)

The proof is based on the same reasoning as the proof of the theorem in [10].
We omit it.

Introduce

$\displaystyle s_{\lambda/\mu}(\mathbf{p}|\mathbf{r})$ $\displaystyle:=$
$\displaystyle\langle
0|\Psi^{*}_{\mu}\hat{\gamma}_{r}(\mathbf{p})\Psi_{\lambda}|0\rangle,$ (5.18)
$\displaystyle Q_{\nu/\theta}({\mathbf{p}^{\rm B}}|\mathbf{r})$
$\displaystyle:=$ $\displaystyle\langle 0|\Phi_{-\theta}\hat{\gamma}_{r}^{\rm
B\pm}(2{\mathbf{p}^{\rm B}})\Phi_{\nu}|0\rangle$ (5.19)

where

$\hat{\gamma}_{r}(\mathbf{p}):=e^{\sum_{j=1}^{\infty}\frac{1}{j}p_{j}\sum_{k\in\mathbf{Z}}\psi_{k}\psi^{\dagger}_{k+j}r(k+1)\cdots
r(k+j)}$ (5.20) $\hat{\gamma}_{r}^{\rm B}(2{\mathbf{p}^{\rm
B}}):=e^{\sum_{j=1,{\rm
odd}}^{\infty}\frac{2}{j}p_{j}\sum_{k\in\mathbf{Z}}(-1)^{k}\phi^{\pm}_{k-j}\phi^{\pm}_{-k}r(k+1)\cdots
r(k+j)}$ (5.21)

By the Wick theorem we obtain

$s_{\lambda/\mu}(\mathbf{p}|\mathbf{r})=\mathrm{det}\left(r(\mu_{i}-i+1)\cdots
r(\lambda_{j}-j)s_{(\lambda_{i}-\mu_{j}-i+j)}(\mathbf{p})\right)_{i,j}$ (5.22)

The similar relation for $Q_{\nu/\theta}({\mathbf{p}^{\rm B}}|\mathbf{r})$ is
more spacious, the Wick theorem yields a Nimmo-type pfaffian formula, we shall
omit it. We have

$s_{\lambda/\mu}(\mathbf{p}^{\prime}|\mathbf{r})=\sum_{(\nu^{+},\nu^{-})\in{\mathcal{P}}(\alpha,\beta)\atop(\theta^{+},\theta^{-})\in{\mathcal{P}}(\hat{\gamma},\delta)}(-1)^{|\theta^{+}|+|\theta^{-}|}a^{\nu^{+},\nu^{-}}_{\alpha,-\beta-1}a^{\theta^{+},\theta^{-}}_{\hat{\gamma},-\delta-1}Q_{\nu^{+}/\theta^{+}}({\mathbf{p}^{\rm
B}}|\mathbf{r})Q_{\nu^{-}/\theta^{-}}({\mathbf{p}^{\rm B}}|\mathbf{r})$ (5.23)

###  5.3 Relation between shifted Schur and shifted projective Schur functions

Let us recall the notion of the shifted Schur function introduced by Okounkov
and Olshanski [28]. In can be defined as

$s^{*}_{\mu}(\lambda)=\frac{\dim\,\lambda/\mu}{\dim\,\lambda}n(n-1)\cdots(n-k+1)=\frac{s_{\lambda/\mu}(\mathbf{p}_{1})}{s_{\lambda}(\mathbf{p}_{1})}$
(5.24)

where $n=|\lambda|$, $k=|\mu|$, $\mathbf{p}_{1}=(1,0,0,\dots)$ and

$\dim\,\lambda/\mu=s_{\lambda/\mu}(\mathbf{p}_{1})(n-k)!,\quad\dim\,\lambda=s_{\lambda}(\mathbf{p}_{1})n!.$

are the number of the standard tableaus of the shape $\lambda/\mu$ and
$\lambda$ respectively, see [17]. The function $s^{*}_{\mu}(\lambda)$ as a
function of the Frobenius coordinates is also known as Frobenius-Schur
function $FS(\alpha,\beta)$

On the other hand, Ivanov [12] introduced the projective analogue of shift
$Q$-functions:

$Q^{*}_{\theta}(\nu)=\frac{Q_{\nu/\theta}(\mathbf{p}_{1})}{Q_{\nu}(\mathbf{p}_{1})}$
(5.25)

Therefore,

$s^{*}_{\mu}(\lambda)s_{\lambda}(\mathbf{p}_{1})=\sum_{(\nu^{+},\nu^{-})\in{\mathcal{P}}(\alpha,\beta)\atop(\theta^{+},\theta^{-})\in{\mathcal{P}}(\hat{\gamma},\delta)}(-1)^{|\theta^{+}|+|\theta^{-}|}a^{\nu^{+},\nu^{-}}_{\alpha,-\beta-1}a^{\theta^{+},\theta^{-}}_{\hat{\gamma},-\delta-1}Q^{*}_{\theta^{+}}(\nu^{+})Q^{*}_{\theta^{-}}(\nu^{-})Q_{\nu^{+}}(\mathbf{p}_{1})Q_{\nu^{-}}(\mathbf{p}_{1})$
(5.26)

Let me add that both $s^{*}$ and $Q^{*}$ were used in the description of the
generalized cut-and-join structure [19],[18] in the topics of Hurwitz and spin
Hurwitz numbers.

Acknowledgements. We are grateful to J. van de Leur for discussions. The work
of J.H. was supported by the Natural Sciences and Engineering Research Council
of Canada (NSERC); that of A. Yu. O. .

###  5.4 Acknowledgements

This work of A.O. is an output of a research project implemented as part of
the Basic Research Program at the National Research University Higher School
of Economics (HSE University).

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##  Appendix A On underlying Lie algebras

The normal ordering of a quadratic elements $\psi_{j}\psi_{k}$ is denoted by
$:\psi_{j}\psi_{k}:$ defined as $:\psi_{j}\psi_{k}:$. As it was noticed in the
works of Kyoto school the elements $E_{j,k}=:\psi_{j}\psi_{k}:$ form the
algebra of infinite matrix with central extention. The normal ordering of

##  Appendix B Appendix. Wick’s theorem

The following summarizes the implication of Wick’s theorem for fermionic VEV’s
in a form that is used repeatedly in this work ( see, e.g., [7], Chapt. 5.1).

######  Theorem B.1 (Wick’s theorem).

The vacuum expectation value of the product of an even number of linear
elements $\\{w_{i}\\}_{1\leq i\leq 2n}$ of the fermionic Clifford algebra is

$\langle 0|w_{1}w_{2}\cdots
w_{2n}|0\rangle=\mathrm{Pf}\left(W_{ij}\right)_{1\leq i,j\leq 2n}$ (B.1)

where $\\{W_{ij}\\}_{1\leq i,j\leq 2n}\\}$ are elements of the skew $2n\times
2n$ matrix defined by

$W_{ij}=\begin{cases}\langle 0|w_{i}w_{j}|0\rangle\quad\ \ \text{ if
}i<j\cr\qquad 0\quad\qquad\ \ \text{ if }i=j\cr-\langle
0|w_{i}w_{j}|0\rangle\quad\text{ if }i>j\end{cases},$ (B.2)

whereas the VEV of the product of an odd number vanishes

$\langle 0|w_{1}w_{2}\cdots w_{2n+1}|0\rangle=0.$ (B.3)

In particular, if half the $w_{i}$’s consist of creation operators
$\\{u_{i}\\}_{i=1,\dots,n}$ and the other half annihilation operators
$\\{v^{\dagger}_{i}\\}_{i=1,\dots,n}$, so that

$\langle 0|u_{i}u_{j}|0\rangle=0,\quad\langle
0|v^{\dagger}_{i}v^{\dagger}_{j}|0\rangle=0,\quad 1\leq i,j\leq n,$ (B.4)

then (B.1) reduces to

$\langle 0|u_{1}v^{\dagger}_{1}\cdots
u_{n}v^{\dagger}_{n}|0\rangle=\mathrm{det}\left(\langle
0|u_{i}v^{\dagger}_{j}|0\rangle\right)_{1\leq i,j\leq n}.$ (B.5)