Multiparameter Schur $Q$-Functions Are Solutions of the BKP Hierarchy

Multiparameter Schur $\boldsymbol{Q}$-Functions
Are Solutions of the BKP Hierarchy††This paper is a contribution to the
Special Issue on Representation Theory and Integrable Systems in honor of
Vitaly Tarasov on the 60th birthday and Alexander Varchenko on the 70th
birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Tarasov-Varchenko.html

Natasha ROZHKOVSKAYA

N. Rozhkovskaya

Department of Mathematics, Kansas State University, Manhattan, KS 66502, USA
<EMAIL_ADDRESS>http://www.math.ksu.edu/~rozhkovs/

Received May 20, 2019, in final form August 23, 2019; Published online August
28, 2019

We prove that multiparameter Schur $Q$-functions, which include as
specializations factorial Schur $Q$-functions and classical Schur
$Q$-functions, provide solutions of the BKP hierarchy.

BKP hierarchy; symmetric functions; factorial Schur $Q$-functions;
multiparameter Schur $Q$-functions; vertex operators

05E05; 17B65; 17B69; 11C20

##  1 Introduction

Integrable systems of the KP (Kadomtsev–Petviashvilli) type hierarchy of
partial differential equations, which corresponds to infinite-dimensional Lie
algebra of type A, and of its type B variant, the BKP hierarchy, have as
solutions renowned families of symmetric functions – Schur polynomials in the
KP case, and Schur $Q$-polynomials in the BKP case [2, 3, 4, 10, 14, 17, 18,
19], etc. In this note we show that multiparameter Schur $Q$-functions also
provide solutions of the BKP hierarchy.

Multiparameter Schur $Q$-functions $Q_{\lambda}^{(a)}$ were introduced and
studied combinatorially in [8]. These symmetric functions are interpolation
analogues of the classical Schur $Q$-functions depending on a sequence of
complex valued parameters $a=(a_{0},a_{1},\dots)$. The definition of
multiparameter Schur $Q$-functions is reproduced in (7.1). Classical Schur
$Q$-functions correspond to $a=(0,0,0,\dots)$, and with the evaluation
$a=(0,1,2,3,\dots)$ the multiparameter Schur $Q$-functions are called
factorial Schur $Q$-functions. These families of symmetric functions proved to
be useful in study of a number of questions of representation theory and
algebraic geometry. Here are a few examples.

The authors of [1, 15, 16] described Capelli polynomials of the queer Lie
superalgebra which form a distinguished family of super-polynomial
differential operators indexed by strict partitions acting on an associative
superalgebra. The eigenvalues of these Capelli polynomials are expressed
through the factorial Schur $Q$-functions.

In [5, 7] the equivariant cohomology of a Lagrangian Grassmannian of a
symplectic or orthogonal types is studied. The restrictions of Schubert
classes to the set of points fixed under the action of a maximal torus of the
symplectic group are calculated in terms of factorial symmetric functions.
Further in [6] factorial Schur $Q$-functions are used to write generators and
relations for the equivariant quantum cohomology rings of the maximal
isotropic Grassmannians of types B, C and D.

In [12] the center of the twisted version of Khovanov’s Heisenberg category is
identified with the algebra generated by classical Schur $Q$-functions
(denoted as $\mathcal{B}_{\rm odd}$ in the exposition below). Factorial Schur
$Q$-functions are described as closed diagrams of this category.

The goal of this note is to show that multiparameter Schur $Q$-functions
$Q_{\lambda}^{(a)}$ are solutions of the BKP hierarchy. The origin for this
phenomena lies in the fact proved in [11] that generating functions of
multiparameter Schur $Q$-functions and of classical Schur $Q$-functions
coincide.

While the BKP hierarchy is described in a wide range of literature on
integrable systems and solitons, for the completeness of exposition and for
the convenience of the reader we formulate the whole setting of the BKP
hierarchy in terms of generating functions of symmetric functions with the
neutral fermions bilinear identity (5.1) as a starting point. We avoid to use
any other facts than the well-known properties of symmetric functions that can
be found in the classical monograph [13], and through the text we provide the
references to the corresponding chapters and examples of that monograph.

It is worth to mention that formulation of the KP and the BKP integrable
systems solely in terms of symmetric functions can be found, e.g., in [9]. The
authors of [9] start with the bilinear identities in integral form, then,
using the Cauchy type orthogonality properties of symmetric functions (cf.
[13, Chapter III, equation (8.13)]), they arrive at Plucker type relations,
and the later ones are transformed into the collection of partial differential
equations of Hirota derivatives that constitute the hierarchy. As it is
mentioned above, our route is traced differently employing the properties of
generating functions of complete, elementary symmetric functions and power
sums. We obtain differential equations of the hierarchy in Hirota form as
coefficients of Taylor expansions. One of the advantages of this approach is
that it directly addresses the corresponding vertex operators actions, since
the later ones are also ‘generating functions’ (formal distributions).

The paper is organized as follows. In Section 2 we recall some facts about
complete, elementary symmetric functions, power sums and classical Schur
$Q$-functions. In Section 3 we describe the action of neutral fermions on the
space generated by classical Schur $Q$-functions. In Section 4 we review
properties of generating functions for multiplication operators and
corresponding adjoint operators and deduce vertex operator form of the formal
distribution of neutral fermions. In Section 5 we review all the steps of
recovering the BKP hierarchy of partial differential equations in Hirota form
from the neutral fermions bilinear identity. In Section 6 we make simple
observation that immediately shows that classical Schur $Q$-functions are
solutions of the BKP hierarchy (which recovers the result of [18]). In Section
7 we introduce multiparameter Schur $Q$-functions, and using the observation
of Section 6, we show that $Q_{\lambda}^{(a)}$ are also solutions of the BKP
hierarchy.

##  2 Schur $\boldsymbol{Q}$-functions

Let $\mathcal{B}$ be the ring of symmetric functions in variables
$(x_{1},x_{2},\dots)$. Consider the families of the following symmetric
functions:

elementary symmetric functions
$\displaystyle\left\\{e_{k}=\sum_{i_{1}<\dots<i_{k}}x_{i_{1}}\cdots
x_{i_{k}}\,|\,{k=0,1,\dots}\right\\},$ complete symmetric functions
$\displaystyle\left\\{h_{k}=\sum_{i_{1}\leq\dots\leq i_{k}}x_{i_{1}}\cdots
x_{i_{k}}\,|\,{k=0,1,\dots}\right\\},$ symmetric power sums
$\displaystyle\left\\{p_{k}=\sum x_{i}^{k}\,|\,{k=0,1,\dots}\right\\}.$

We set $e_{k}=h_{k}=0$ for $k<0$. It is well-known [13, Chapter I.2], that
each of these families generate $\mathcal{B}$ as a polynomial ring:

$\displaystyle\mathcal{B}={\mathbb{C}}[p_{1},p_{2},p_{3},\dots]={\mathbb{C}}[e_{1},e_{2},e_{3},\dots]={\mathbb{C}}[h_{1},h_{2},h_{3},\dots].$

Combine the families $h_{k}$, $e_{k}$, $p_{k}$ into generating functions

$\displaystyle H(u)=\sum_{k\geq 0}\frac{h_{k}}{u^{k}},\qquad E(u)=\sum_{k\geq
0}\frac{e_{k}}{u^{k}},\qquad P(u)=\sum_{k\geq 1}\frac{p_{k}}{u^{k}}.$

The following facts are well-known [13, Chapter I.2].

######  Lemma 2.1.

$\displaystyle H(u)=\prod_{i}\frac{1}{1-x_{i}/u},\qquad
E(u)=\prod_{i}{1+x_{i}/u},\qquad E(-u)H(u)=1,$ $\displaystyle
H(u)=\exp\left(\sum_{n\geq 1}\frac{1}{n}\frac{p_{n}}{u^{n}}\right),\qquad
E(u)=\exp\left(\sum_{n\geq 1}\frac{(-1)^{n-1}}{n}\frac{p_{n}}{u^{n}}\right).$

We introduce one more family of symmetric functions
$\\{Q_{k}=Q_{k}(x_{1},x_{2},\dots)\\}$ with $(k=0,1,\dots)$ as the
coefficients of the generating function

$\displaystyle Q(u)=\sum_{k\geq
0}\frac{Q_{k}}{u^{k}}=\prod_{i}\frac{u+x_{i}}{u-x_{i}}.$ (2.1)

From Lemma 2.1 and (2.1) we immediately get relations of the next lemma.

######  Lemma 2.2.

$\displaystyle Q(u)=E(u)H(u)=R(u)^{2},\qquad
R(u)=\exp\left(\sum_{n\in{\mathbb{N}}_{\rm odd}}\frac{p_{n}}{nu^{n}}\right),$

where ${\mathbb{N}}_{\rm odd}=\\{1,3,5,\dots\\}$.

Note that $Q(u)Q(-u)=1$, which implies that $Q_{r}$ with even $r$ can be
expressed algebraically through $Q_{r}$ with odd $r$:

$\displaystyle
Q_{2m}=\sum_{r=1}^{m-1}(-1)^{r-1}Q_{r}Q_{2m-r}+\frac{1}{2}(-1)^{m-1}Q^{2}_{m}.$

More generally, Schur $Q$-functions $Q_{\lambda}$ labeled by strict partitions
are defined as a specialization of Hall–Littlewood polynomials [13, Chapter
III.2].

######  Definition 2.3.

Let $\lambda=(\lambda_{1}>\lambda_{2}>\dots>\lambda_{l})$ be a strict
partition. Let $l\leq N$. Schur $Q$-polynomial
$Q_{\lambda}(x_{1},\dots,x_{N})$ is the symmetric polynomial in variables
$x_{i}$’s defined by the formula

$\displaystyle
Q_{\lambda}(x_{1},\dots,x_{N})=\frac{2^{l}}{(N-l)!}\sum_{\sigma\in
S_{N}}\prod_{i=1}^{l}x_{\sigma(i)}^{\lambda_{i}}\prod_{i<j}\frac{x_{\sigma(i)}+x_{\sigma(j)}}{x_{\sigma(i)}-x_{\sigma(j)}}.$
(2.2)

Alternatively, Schur $Q$-polynomial
$Q_{\lambda}=Q_{\lambda}(x_{1},\dots,x_{N})$ for $N>l$ is the coefficient of
${u^{-\lambda_{1}}\cdots u^{-\lambda_{l}}}$ in the formal generating function

$\displaystyle
Q(u_{1},\dots,u_{l})=\sum_{\lambda_{1},\dots,\lambda_{l}\in{\mathbb{Z}}}\frac{Q_{\lambda}}{u^{\lambda_{1}}\cdots
u^{\lambda_{l}}}=\prod_{1\leq i<j\leq
l}\frac{u_{j}-u_{i}}{u_{j}+u_{i}}\prod_{i=1}^{l}Q(u_{i}),$ (2.3)

where it is understood that

$\displaystyle\frac{u_{j}-u_{i}}{u_{j}+u_{i}}=1+2\sum_{r\geq
1}(-1)^{r}u_{i}^{r}u_{j}^{-r},$

and $Q(u)$ is given by (2.1) [13, Chapter III, equation (8.8)]. Schur
$Q$-polynomials have a stabilization property, hence, one can omit the number
$N$ of variables $x_{i}^{\prime}s$ as long as it is not less than the length
of the partition $\lambda$ and consider $Q_{\lambda}$ as functions of
infinitely many variables $(x_{1},x_{2},\dots)$.

##  3 Action of neutral fermions on bosonic space
$\boldsymbol{\mathcal{B}_{\rm odd}}$

Consider the subalgebra $\mathcal{B}_{\rm odd}$ of $\mathcal{B}$ generated by
odd ordinary Schur $Q$-functions: $\mathcal{B}_{\rm
odd}={\mathbb{C}}[Q_{1},Q_{3},\dots]$. It is known that $\mathcal{B}_{\rm
odd}$ is also a polynomial algebra in odd power sums $\mathcal{B}_{\rm
odd}={\mathbb{C}}[p_{1},p_{3},\dots]$ and that Schur $Q$-functions
$Q_{\lambda}$ labeled by strict partitions constitute a linear basis of
$\mathcal{B}_{\rm odd}$ [13, Chapter III.8, equation (8.9)].

Define operators $\\{\varphi_{k}\\}_{k\in{\mathbb{Z}}}$ acting on the
coefficients of generating functions $Q(u_{1},\dots,u_{l})$ by the rule

$\displaystyle\Phi(v)Q(u_{1},\dots,u_{l})=Q(v,u_{1},\dots,u_{l})$ (3.1)

with $\Phi(v)=\sum\limits_{m\in{\mathbb{Z}}}{\varphi_{m}}{v^{-m}}$. Then in
the expansion (2.3)

$\displaystyle\varphi_{m}\colon\ Q_{\lambda}\mapsto Q_{(m,\lambda)}.$

Observe that from (2.3)

$\displaystyle(\Phi(u)\Phi(v)+\Phi(v)\Phi(u))Q(u_{1},\dots,u_{l})$
$\displaystyle\qquad{}=2\left(1+\sum_{r\geq 1}(-1)^{r}u^{r}v^{-r}+\sum_{r\geq
1}(-1)^{r}v^{r}u^{-r}\right)A(u,v,u_{1},\dots,u_{l})$
$\displaystyle\qquad{}=2\sum_{r\in{\mathbb{Z}}}\left(\frac{-u}{v}\right)^{r}A(u,v,u_{1},\dots,u_{l}),$

where

$\displaystyle A(u,v,u_{1},\dots,u_{l})=\prod_{1\leq j\leq
l}\frac{(u_{j}-u)(u_{j}-v)}{(u_{j}+u)(u_{j}+v)}Q(u)Q(v)Q(u_{1},\dots,u_{l}).$

Using that $\delta(u,v)=\sum\limits_{r\in{\mathbb{Z}}}{u^{r}}{v^{-(r+1)}}$ is
a formal delta distribution with the property
$\delta(u,v)a(u)=\delta(u,v)a(v)$ for any formal distribution
$a(u)=\sum\limits_{n\in{\mathbb{Z}}}a_{n}u^{n}$, and that $Q(u)Q(-u)=1$, we
get

$\displaystyle\left(\Phi(u)\Phi(v)+\Phi(v)\Phi(u)\right)Q(u_{1},\dots,u_{l})=2v\delta(-u,v)Q(u_{1},\dots,u_{l}).$
(3.2)

Since coefficients of the expansion of $Q(u_{1},\dots,u_{l})$ in powers of
$u_{1},\dots,u_{l}$ include Schur $Q$-functions $Q_{\lambda}$, and the latter
form a linear basis of $\mathcal{B}_{\rm odd}$, it follows that (3.1) provides
the action of well-defined operators $\\{\varphi_{k}\\}_{k\in{\mathbb{Z}}}$ on
$\mathcal{B}_{\rm odd}$:

$\displaystyle\varphi_{k}(Q_{\lambda})=Q_{(k,\lambda)}.$

Relation (3.2) on generating functions is equivalent to the commutation
relations

$\displaystyle[\varphi_{m},\varphi_{n}]_{+}=2(-1)^{m}\delta_{m+n,0}\qquad\text{for}\quad
m,n\in{\mathbb{Z}}.$ (3.3)

Thus, operators $\\{\varphi_{i}\\}_{i\in{\mathbb{Z}}}$ and ${1}$ provide the
action of Clifford algebra ${\rm Cl}_{\varphi}$ of neutral fermions on the
Fock space $\mathcal{B}_{\rm odd}$. Note that for any strict partition
$\lambda=(\lambda_{1}>\lambda_{2}>\dots>\lambda_{l})$

$\displaystyle
Q_{(\lambda_{1},\dots\lambda_{l})}=\varphi_{\lambda_{1}}\cdots\varphi_{\lambda_{l}}(1),$
(3.4)

or in terms of generating functions,

$\displaystyle Q(u_{1},\dots,u_{l})=\Phi(u_{1})\cdots\Phi(u_{l})(1).$ (3.5)

Formulae (3.4), (3.5) sometimes are called the vertex operator realization of
Schur $Q$-functions.

##  4 Vertex operator form of formal distribution
of neutral fermions

It will be convenient for us to consider $\mathcal{B}_{\rm odd}$ as a subring
of the ring of symmetric functions $\mathcal{B}$. This allows us to recover
the celebrated vertex operator form of the formal distribution of neutral
fermions $\Phi(u)$ from no-less celebrated properties of generating functions
of complete and elementary symmetric functions. All of these properties are
discussed in [13, Chapter I].

The ring of symmetric functions $\mathcal{B}$ possesses a bilinear form
$(\cdot,\cdot)$ [13, Chapter I, equation (4.5)] defined on the linear basis of
monomials of power sums labeled by partitions $\lambda$ and $\mu$ as

$\displaystyle(p_{\lambda_{1}}\cdots p_{\lambda_{l}},p_{\mu_{1}}\cdots
p_{\mu_{l}})=z_{\lambda}\delta_{\lambda,\mu},$

where $z_{\lambda}=\prod i^{m_{i}}m_{i}!$ and $m_{i}=m_{i}(\lambda)$ is the
number of parts of $\lambda$ equal to $i$.

We will use this form and its restriction to $\mathcal{B}_{\rm odd}$ to define
adjoint operators111Traditionally, one uses rescaled form on $\mathcal{B}_{\rm
odd}$ defined as
$(p_{\lambda},p_{\mu})=2^{-l(\lambda)}z_{\lambda}\delta_{\lambda,\mu},$ where
$l(\lambda)$ is the number of parts of $\lambda$, but rescaling is not
necessary for our purposes, since in the rescaled form
$p_{n}^{\perp}=n/2\cdot\partial/\partial{p_{n}}$ (see [13, Chapter III.8,
Example 11]). of the multiplication operators. By definition, given an element
$f\in\mathcal{B}$, the operator $f^{\perp}$ adjoint to the operator of
multiplication by $f$ is given by the rule

$\displaystyle\big{(}f^{\perp}g,h\big{)}=(g,fh)\qquad\text{for any}\quad
g,h\in\mathcal{B}.$

[13, Chapter I.5, Example 3] contains the following statement. Consider a
symmetric function $f=f(p_{1},p_{2},\dots)$ expressed as a polynomial in power
sums $p_{i}$. Then the adjoint operator on $\mathcal{B}$ to the multiplication
operator by $f$ is given by

$\displaystyle
f^{\perp}=f\left(\frac{\partial}{\partial{p_{1}}},\frac{2\partial}{\partial{p_{2}}},\frac{3\partial}{\partial{p_{3}}},\dots\right).$
(4.1)

In particular $p_{n}^{\perp}=n\partial/\partial{p_{n}}$.

Combine the corresponding adjoint operators of the families $h_{k}$, $e_{k}$,
$p_{k}$ and $Q_{k}$ into generating functions

$\displaystyle H^{\perp}(u)=\sum_{k\geq 0}{h_{k}^{\perp}}{u^{k}},\qquad
E^{\perp}(u)=\sum_{k\geq 0}{e_{k}^{\perp}}{u^{k}},$ $\displaystyle
P^{\perp}(u)=\sum_{k\geq 1}{p_{k}^{\perp}}{u_{k}},\qquad
Q^{\perp}(u)=\sum_{k\geq 0}{Q_{k}^{\perp}}{u_{k}}.$

Then (4.1) immediately implies the following relations.

######  Lemma 4.1.

$\displaystyle H^{\perp}(u)=\exp\left(\sum_{n\geq 1}\frac{\partial}{\partial
p_{n}}{u^{n}}\right),\qquad E^{\perp}(u)=\exp\left(\sum_{n\geq
1}{(-1)^{n-1}}\frac{\partial}{\partial p_{n}}{u^{n}}\right),$ $\displaystyle
Q^{\perp}(u)=E^{\perp}(u)H^{\perp}(u)=R^{\perp}(u)^{2},\qquad
R^{\perp}(u)=\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}\frac{\partial}{\partial p_{n}}{u^{n}}\right),$

where ${\mathbb{N}}_{\rm odd}=\\{1,3,5,\dots\\}$.

The proof of the next lemma is outlined in [13, Chapter I.5, Example 29].

######  Lemma 4.2.

The following commutation relations on generating functions of multiplication
and adjoint operators acting on $\mathcal{B}$ hold:

$\displaystyle H^{\perp}(u)\circ H(v)=(1-u/v)^{-1}H(u)\circ H^{\perp}(v),$
$\displaystyle H^{\perp}(u)\circ E(v)=(1+u/v)E(u)\circ H^{\perp}(v),$
$\displaystyle E^{\perp}(u)\circ H(v)=(1+u/v)H(u)\circ E^{\perp}(v),$
$\displaystyle E^{\perp}(u)\circ E(v)=(1-u/v)^{-1}E(u)\circ E^{\perp}(v).$

######  Corollary 4.3.

$\displaystyle H^{\perp}(u)\circ Q(v)=\frac{v+u}{v-u}Q(u)\circ H^{\perp}(v),$
$\displaystyle E^{\perp}(u)\circ Q(v)=\frac{v+u}{v-u}Q(u)\circ E^{\perp}(v),$
$\displaystyle R^{\perp}(u)\circ Q(v)=\frac{v+u}{v-u}Q(u)\circ R^{\perp}(v).$

###### Proof.

For the first and second one we use that $Q(u)=E(u)H(u)$. Observe that

$\displaystyle H^{\perp}(u)|_{\mathcal{B}_{\rm
odd}}=E^{\perp}(u)|_{\mathcal{B}_{\rm odd}}=R^{\perp}(u)|_{\mathcal{B}_{\rm
odd}}.$

In other words, since $Q_{k}$ does not depend on even power sums $p_{2r}$, we
can add terms ${\partial}/{\partial p_{2r}}$ in the sum under the exponent
when applying to elements of $\mathcal{B}_{\rm odd}$:

$\displaystyle R(u)^{\perp}(Q(v))=\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}\frac{\partial}{\partial p_{n}}{u^{n}}\right)Q(v)=\exp\left(\sum_{n\geq
1}\frac{\partial}{\partial p_{n}}{u^{n}}\right)Q(v)=H^{\perp}(u)Q(v).$ ∎

We arrive at the vertex operator form of formal distribution of neutral
fermions.

######  Proposition 4.4.

$\displaystyle\Phi(v)=Q(v)R(-v)^{\perp}=\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}\frac{2p_{n}}{n}\frac{1}{v^{n}}\right)\exp\left(-\sum_{n\in{\mathbb{N}}_{\rm
odd}}\frac{\partial}{\partial p_{n}}{v^{n}}\right).$ (4.2)

###### Proof.

From Corollary 4.3, the action of the operator $Q(v)R(-v)^{\perp}$ on the
coefficients of generating function $Q(u_{1},\dots,u_{l})$ coincides with the
action of $\Phi(v)$:

$\displaystyle Q(v)R(-v)^{\perp}(Q(u_{1},\dots,u_{l}))=Q(v)\prod_{1\leq
i<j\leq
l}\frac{u_{j}-u_{i}}{u_{j}+u_{i}}\,R(-v)^{\perp}\left(\prod_{i=1}^{l}Q(u_{i})\right)$
$\displaystyle\qquad{}=Q(v)\prod_{1\leq j<i\leq
l}\frac{u_{j}-u_{i}}{u_{j}+u_{i}}\prod_{i=1}^{l}\frac{v-u_{i}}{v+u_{i}}\prod_{i=1}^{l}Q(u_{i})=Q(v,u_{1},\dots,u_{l}).$

Since coefficients of $Q(u_{1},\dots,u_{l})$ contain a linear basis of
$\mathcal{B}_{\rm odd}$, the equality (4.2) follows. ∎

##  5 The neutral fermions bilinear identity

Let

$\displaystyle\Omega=\sum_{n}\varphi_{n}\otimes(-1)^{n}\varphi_{-n}.$

One looks for the solutions in $\mathcal{B}_{\rm odd}$ of the neutral fermions
bilinear identity

$\displaystyle\Omega(\tau\otimes\tau)=\tau\otimes\tau,$ (5.1)

where $\tau=\tau(\tilde{p})=\tau(2p_{1},2p_{3}/3,2p_{5}/5,\dots)$. It is known
[2, 3, 4, 10] that (5.1) is equivalent to an infinite integrable system of
partial differential equations called the BKP hierarchy. Further in Section 6
a simple observation explains, why Schur $Q$-functions constitute solutions of
the neutral fermions bilinear identity, and hence of the BKP hierarchy.

In this section we would like to make a small deviation and review the steps
of recovering the BKP hierarchy of partial differential equations in the
Hirota form from the neutral fermions bilinear identity. This is certainly a
well-known procedure. However, the explicit calculations are often omitted in
the literature, and we would like to provide them here for the convenience of
the reader.

Note that $\Omega$ is the constant coefficient of the formal distribution
$\Phi(u)\otimes\Phi(-u)$, or, in terms of residue,

$\displaystyle\Omega=\mathop{\operatorname{Res}}\limits_{u=0}\frac{1}{u}\Phi(u)\otimes\Phi(-u).$
(5.2)

We identify $\mathcal{B}_{\rm odd}\otimes\mathcal{B}_{\rm odd}$ with
${\mathbb{C}}[p_{1},p_{3},\dots]\otimes{\mathbb{C}}[r_{1},r_{3},\dots]$ – two
copies of polynomial rings, where variables in each of them play role of power
sum symmetric functions. Set $\tilde{p}=(2p_{1},2p_{3}/3,\allowbreak
2p_{5}/5,\dots)$, $\tilde{r}=(2r_{1},2r_{3}/3,2r_{5}/5,\dots)$. Then
$\partial_{p_{n}}=2\partial_{\tilde{p}_{n}}/n$ and

$\displaystyle\Phi(u)\tau\otimes\Phi(-u)\tau=\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}(\tilde{p}_{n}-\tilde{r}_{n})\frac{1}{u^{n}}\right)$
$\displaystyle\hphantom{\Phi(u)\tau\otimes\Phi(-u)\tau=}{}\times\exp\left(-\sum_{n\in{\mathbb{N}}_{\rm
odd}}\frac{2}{n}\left(\frac{\partial}{\partial\tilde{p}_{n}}-\frac{\partial}{\partial\tilde{r}_{n}}\right){u^{n}}\right)\tau(\tilde{p})\tau(\tilde{r}).$

Introduce the change of variables

$\displaystyle\tilde{p}_{n}=x_{n}-y_{n},\qquad\tilde{r}_{n}=x_{n}+y_{n}.$

Then

$\displaystyle\Phi(u)\tau\otimes\Phi(-u)\tau=\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}-2y_{n}\frac{1}{u^{n}}\right)\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}\frac{2}{n}\frac{\partial}{\partial y_{n}}u^{n}\right)\tau(x-y)\tau(x+y)$

with $(x\pm y)=(x_{1}\pm y_{1},x_{3}\pm y_{3},x_{5}\pm y_{5},\dots)$.

######  Definition 5.1.

Let $P(D)$ be a multivariable polynomial in the collection of variables
$D=(D_{1},D_{2},\dots)$, let $f(x)$, $g(x)$ be differentiable functions in
$x=(x_{1},x_{2},\dots)$.

The Hirota derivative $P(D)f\cdot g$ is a function in variables
$(x_{1},x_{2},\dots)$ given by the expression

$\displaystyle P(D)f\cdot
g=P(\partial_{z_{1}},\partial_{z_{1}},\dots)f(x+z)g(x-z)|_{z=0},$

where $x\pm z=(x_{1}\pm z_{1},x_{2}\pm z_{2},\dots)$.

For example,

$\displaystyle D_{i}^{n}f\cdot g=\sum_{k=0}^{n}(-1)^{k}{n\choose
k}\frac{\partial^{k}f}{\partial x_{i}^{k}}\frac{\partial^{n-k}g}{\partial
x_{i}^{n-k}},$

which implies in particular that odd Hirota derivatives are tautologically
zero when $f=g$:

$\displaystyle D_{i}^{2n+1}f\cdot f=0\qquad\text{for}\quad n=0,1,2,\dots.$

The following lemma allows one to rewrite bilinear identity (5.1) in terms of
the Hirota derivatives.

######  Lemma 5.2.

$\displaystyle\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}\frac{2}{n}\frac{\partial}{\partial
y_{n}}{u^{n}}\right)\tau(x-y)\tau(x+y)=\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}\left(y_{n}+\frac{2}{n}u^{n}\right)D_{n}\right)\tau\cdot\tau.$

###### Proof.

By the Taylor series expansion,

$\displaystyle{\rm
e}^{a\partial/\partial_{y}}g(y)=\sum_{n=0}^{\infty}\frac{a^{n}g^{(n)}(y)}{n!}=g(y+a).$
(5.3)

Applying (5.3) twice with $t=(t_{1},t_{3},t_{5},\dots)$,
$\tilde{u}=\big{(}2u,2u^{3}/3,2u^{5}/5,\dots\big{)}$,

$\displaystyle\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}\frac{2}{n}\frac{\partial}{\partial
y_{n}}u^{n}\right)\tau(x-y)\tau(x+y)=\tau(x+y+\tilde{u})\tau(x-y-\tilde{u})$
$\displaystyle\qquad{}=\tau(x+y+\tilde{u}+t)\tau(x-(y+\tilde{u}+t))|_{t=0}$
$\displaystyle\qquad{}=\left.\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}\left(y_{n}+\frac{2}{n}u^{n}\right)\frac{\partial}{\partial
t_{n}}\right)\tau(x+t)\tau(x-t)\right|_{t=0}.$ ∎

Thus, we can write in terms of Hirota derivatives

$\displaystyle\Phi(u)\tau\otimes\Phi(-u)\tau=\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}\frac{-2y_{n}}{u^{n}}\right)$
$\displaystyle\hphantom{\Phi(u)\tau\otimes\Phi(-u)\tau=}{}\times\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}\frac{2}{n}D_{n}u^{n}\right)\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}y_{n}D_{n}\right)\tau\cdot\tau.$ (5.4)

In order to compute
$\mathop{\operatorname{Res}}\limits_{u=0}\frac{1}{u}\Phi(u)\tau\otimes\Phi(-u)\tau$,
which is just the coefficient of $u^{0}$ of $\Phi(u)\tau\otimes\Phi(-u)\tau$,
we recall the following well-known facts on the composition of exponential
series with generating series. Their proofs can be done, e.g., by induction,
or again found in [13, Chapter I].

######  Proposition 5.3.

Let $S(u)=\sum\limits_{k=0}^{\infty}S_{k}\frac{1}{u^{k}}$ and
$X(u)=\sum\limits_{k=1}^{\infty}X_{k}\frac{1}{u^{k}}$ be related by

$\displaystyle\exp(X(u))=S(u).$

Then the following statements hold

$\displaystyle
S_{k}=\sum_{s=1}^{k}\sum_{l_{1}+2l_{2}+\dots+sl_{s}=k,\,l_{i}\geq
1}\frac{1}{l_{1}!\cdots l_{s}!}X_{1}^{l_{1}}\cdots X_{s}^{l_{s}},$
$\displaystyle S_{k}=\det\frac{1}{n!}\begin{pmatrix}X_{1}&-1&0&0&\dots&0\\\
2X_{2}&X_{1}&-2&0&\dots&0\\\ 3X_{3}&2X_{2}&X_{1}&-3&\dots&0\\\
\dots&\dots&\dots&\dots&\dots&0\\\
kX_{k}&(k-1)X_{k-1}&(k-2)X_{k-2}&(k-3)X_{k-3}&\dots&X_{1}\\\ \end{pmatrix},$
$\displaystyle
X_{k}=\frac{(-1)^{k-1}}{k}\det\begin{pmatrix}S_{1}&1&0&0&\dots&0\\\
2S_{2}&S_{1}&1&0&\dots&0\\\ 3S_{3}&S_{2}&S_{1}&1&\dots&0\\\
\dots&\dots&\dots&\dots&\dots&0\\\
kS_{k}&S_{k-1}&S_{k-2}&S_{k-3}&\dots&S_{1}\\\ \end{pmatrix}.$

######  Example 5.4.

$\displaystyle S_{0}=1,\qquad S_{1}=X_{1},\qquad
S_{2}=\frac{1}{2}X_{1}^{2}+X_{2},\qquad
S_{3}=S_{3}+X_{2}X_{1}+\frac{1}{6}X_{1}^{3},$ $\displaystyle
S_{4}=X_{4}+X_{3}X_{1}+\frac{1}{2}X_{2}^{2}+\frac{1}{2}X_{2}X_{1}^{2}+\frac{1}{24}X_{1}^{4}.$

By Lemma 2.1, when $X$ variables in these formulae are interpreted as
normalized power sums $X_{k}=p_{k}/k$, $S_{k}$’s are identified with complete
symmetric functions $h_{k}$’s.

######  Example 5.5.

Let $X_{2k}=0$ for $k=1,2,\dots.$ Then the first few
$S_{n}=S_{n}(X_{1},0,X_{3},\dots)$ are given by

$\displaystyle S_{0}=1,\qquad S_{1}=X_{1},\qquad
S_{2}=\frac{1}{2}X_{1}^{2},\qquad S_{3}=X_{3}+\frac{1}{6}X_{1}^{3},\qquad
S_{4}=X_{3}X_{1}+\frac{1}{24}X_{1}^{4},$ $\displaystyle
S_{5}=\frac{1}{120}X_{1}^{5}+\frac{1}{2}X_{1}^{2}X_{3}+X_{5},\qquad
S_{6}=\frac{1}{720}X_{1}^{6}+\frac{1}{6}X_{1}^{3}X_{3}+\frac{1}{2}X_{3}^{2}+X_{1}X_{5},$
$\displaystyle
S_{7}=\frac{1}{5040}X_{1}^{7}+\frac{1}{24}X_{1}^{4}X_{3}+\frac{1}{2}X_{1}X_{3}^{2}+\frac{1}{2}X_{1}X_{5}+X_{7}.$

Note that by Lemma 2.2 when $X$ variables in these formulae are interpreted as
odd normalized power sums $X_{2k+1}=2p_{2k+1}/(2k+1)$, $S_{k}$’s are
identified with Schur $Q$-functions $Q_{k}$’s.

Using the statement of Proposition 5.3, we can write the coefficient of
$u^{0}$ of (5.4) as

$\displaystyle\sum_{m=0}^{\infty}S_{m}\left(\tilde{y}\right)S_{m}\big{(}\tilde{D}\big{)}\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}y_{n}D_{n}\right)\tau\cdot\tau=\tau(x-y)\cdot\tau(x+y),$ (5.5)

where $\tilde{y}=({-2y_{1}},0,-2y_{3},\dots)$,
$\tilde{D}=(2D_{1},0,2D_{3}/3,0,\dots)$.

Note that $S_{0}=1$ and $\exp\Big{(}\sum\limits_{n\in{\mathbb{N}}_{\rm
odd}}y_{n}D_{n}\Big{)}\tau\cdot\tau=\tau(x-y)\cdot\tau(x+y)$, hence we can
rewrite (5.5) as

$\displaystyle\sum_{m=1}^{\infty}S_{m}(\tilde{y})S_{m}\big{(}\tilde{D}\big{)}\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}y_{n}D_{n}\right)\tau\cdot\tau=0.$ (5.6)

To obtain the equations of the BKP hierarchy, one expands the left hand side
of (5.6) in monomials $y_{1}^{m_{1}}y_{2}^{m_{2}}\cdots y_{N}^{m_{N}}$ to
obtain as coefficients Hirota operators in terms of $D_{k}$’s.

For example, let us compute the coefficient of $y_{3}^{2}$. In the expansion
of

$\displaystyle\big{(}S_{1}(\tilde{y})S_{1}\big{(}\tilde{D}\big{)}+S_{2}(\tilde{y})S_{2}\big{(}\tilde{D}\big{)}+S_{3}(\tilde{y})S_{3}\big{(}\tilde{D}\big{)}+\cdots\big{)}\left(1+\sum
y_{i}D_{i}+\frac{1}{2}\left(\sum y_{i}D_{i}\right)^{2}+\cdots\right)$

the term $y_{3}^{2}$ appears in
$S_{3}(\tilde{y})S_{3}\big{(}\tilde{D}\big{)}\times y_{3}D_{3}$ and in
$S_{6}(\tilde{y})S_{6}\big{(}\tilde{D}\big{)}\times 1$. Using the expansions
of Example 5.5, the coefficient of $y_{3}^{2}$ is

$\displaystyle-2S_{3}\big{(}\tilde{D}\big{)}D_{3}+2S_{6}\big{(}\tilde{D}\big{)}=\frac{8}{45}\big{(}D_{1}^{6}-5D_{1}D_{3}-5D_{3}^{2}+9D_{1}D_{5}\big{)},$

which provides the Hirota bilinear form of the BKP equation that gives the
name to the hierarchy

$\displaystyle\big{(}D_{1}^{6}-5D_{1}D_{3}-5D_{3}^{2}+9D_{1}D_{5}\big{)}\tau\cdot\tau=0.$

######  Remark 5.6.

Writing the residue (5.2) as a contour integral, one gets the BKP in its
integral form

$\displaystyle\oint\frac{1}{2\pi{\rm i}u}\exp\left(\sum_{n\in{\mathbb{N}}_{\rm
odd}}(\tilde{p}_{n}-\tilde{r}_{n})\frac{1}{u^{n}}\right)\exp\left(-\sum_{n\in{\mathbb{N}}_{\rm
odd}}\frac{2}{n}\left(\frac{\partial}{\partial\tilde{p}_{n}}-\frac{\partial}{\partial\tilde{r}_{n}}\right){u^{n}}\right)\tau(\tilde{p})\tau(\tilde{r})$
$\displaystyle\qquad{}=\tau(\tilde{p})\tau(\tilde{r}).$

##  6 Commutation relation for the bilinear identity

Our goal is to show that multiparameter Schur $Q$-functions are solutions of
the neutral fermions bilinear identity (5.1), thus they provide solutions of
the BKP hierarchy.

Let $X=\sum\limits_{n>0}A_{n}\varphi_{n}$ for some $A_{n}\in{\mathbb{C}}$.
From (3.3) one gets $X^{2}=0$.

######  Proposition 6.1.

$\displaystyle\Omega(X\otimes X)=(X\otimes X)\Omega.$

###### Proof.

$\displaystyle\Omega(X\otimes X)$
$\displaystyle=\sum_{k\in{\mathbb{Z}}}\varphi_{k}X\otimes(-1)^{k}\varphi_{-k}X$
$\displaystyle=\sum_{k\in{\mathbb{Z}}}(-X\varphi_{k}+[\varphi_{k},X]_{+})\otimes(-1)^{k}(-X\varphi_{-k}+[\varphi_{-k},X]_{+}).$

Note that
$[\varphi_{k},X]_{+}=\big{[}\varphi_{k},\sum\limits_{n>0}A_{n}\varphi_{n}\big{]}_{+}=2\sum\limits_{n>0}(-1)^{n}A_{n}\delta_{k+n,0}$,
hence

$\displaystyle\Omega(X\otimes X)=(X\otimes
X)\Omega-\sum_{n>0}2(-1)^{n}A_{n}\otimes(-1)^{n}X\varphi_{n}-\sum_{n>0}(-1)^{n}X\varphi_{n}\otimes
2(-1)^{n}A_{n}$ $\displaystyle\hphantom{\Omega(X\otimes
X)=}{}+4\sum_{k\in{\mathbb{Z}}}\sum_{m,n>0}(-1)^{n}A_{n}\delta_{n+k,0}\otimes(-1)^{m}A_{m}\delta_{m-k,0}$
$\displaystyle\hphantom{\Omega(X\otimes X)}{}=(X\otimes X)\Omega-2\otimes
X^{2}-X^{2}\otimes
2+4\sum_{m,n>0}(-1)^{n}A_{n}\otimes(-1)^{m}A_{m}\delta_{m+n,0}.$

We use that $X^{2}=0$, and since both $m$ and $n$ in the last sum are always
positive, the last term is also zero. ∎

######  Corollary 6.2.

Let $\tau\in\mathcal{B}_{\rm odd}$ be a solution of (5.1), and let
$X=\sum\limits_{n>0}A_{n}\varphi_{n}$ with $A_{n}\in{\mathbb{C}}$. Then
$\tau^{\prime}=X\tau$ is also a solution of (5.1).

The vertex operator presentation (3.4) of Schur $Q$-functions and Corollary
6.2 immediately imply that Schur $Q$-functions are solutions of (5.1), since
constant function $1$ is a solution of (5.1). This argument reproves the
result of [18] and easily extends to more general case of multiparameter Schur
$Q$-functions defined in the next section.

##  7 Multiparameter Schur $\boldsymbol{Q}$-functions are solutions
of the BKP hierarchy

Multiparameter Schur $Q$-functions were introduced in [8] as a generalization
of definition (2.2). Fix an infinite sequence of complex numbers
$a=(a_{0},a_{1},a_{2},\dots)$. Consider the analogue of a power of a variable
$x$

$\displaystyle(x|a)^{k}=(x-a_{0})(x-a_{2})\cdots(x-a_{k-1}).$

We also define a shift operation $\tau\colon a_{k}\mapsto a_{k+1}$, so that

$\displaystyle(x|\tau a)^{k}=(x-a_{1})(x-a_{2})\cdots(x-a_{k}).$

######  Definition 7.1.

Let $\alpha=(\alpha_{1},\dots,\alpha_{l})$ be a vector with non-negative
integer coefficients $\alpha_{i}\in{\mathbb{Z}}_{\geq 0}$. The multiparameter
Schur $Q$-function in variables $(x_{1},\dots,x_{N})$ with $l\leq N$ is
defined by

$\displaystyle
Q^{(a)}_{\alpha}(x_{1},\dots,x_{N})=\frac{2^{l}}{(N-l)!}\sum_{\sigma\in
S_{N}}\prod_{i=1}^{l}(x_{\sigma(i)}|a)^{\alpha_{i}}\prod_{i\leq l,i<j\leq
N}\frac{x_{\sigma(i)}+x_{\sigma(j)}}{x_{\sigma(i)}-x_{\sigma(j)}}.$ (7.1)

When $a=(0,0,0,\dots)$ and $\alpha$ is a strict partition, one gets back
(2.2), the classical Schur $Q$-functions $Q_{\alpha}(x_{1},\dots,x_{N})$. The
evaluation $a=(0,1,2,\dots)$ gives factorial Schur $Q$-functions denoted as
$Q^{*}_{\alpha}(x)$, those applications are outlined in the introduction. The
multiparameter Schur $Q$-functions enjoy a stability property, hence one can
consider $Q^{(a)}_{\alpha}(x_{1},x_{2},\dots)$ to be a function of infinitely
many variables.

Note from (7.1) that for any permutation $\sigma\in S_{l}$,

$\displaystyle
Q^{(a)}_{\alpha}(x_{1},\dots,x_{N})=(-1)^{\sigma}Q^{(a)}_{\sigma(\alpha)}(x_{1},\dots,x_{N}),$
(7.2)

where $(-1)^{\sigma}$ is the sign of permutation $\sigma$ [11, Proposition 3].
Hence, $Q^{(a)}_{\alpha}=0$ if $\alpha_{i}=\alpha_{j}$ for some $i$, $j$, and
for a vector $\alpha=(\alpha_{1},\dots,\alpha_{l})$ with positive distinct
integer coefficients $\alpha_{i}\in{\mathbb{Z}}_{>0}$, function
$Q^{(a)}_{\alpha}$ coincides up to a sign with another
$Q^{(a)}_{\alpha^{\prime}}$ labeled by strict partition $\alpha^{\prime}$.

One can check directly the following transitions between regular and
multiparameter powers of variables.

######  Lemma 7.2.

For $n=0,1,2,\dots$

$\displaystyle(u-a_{1})\cdots(u-a_{n})=\sum_{k=0}^{\infty}(-1)^{n-k}e_{n-k}(a_{1},\dots,a_{n})u^{k},$
$\displaystyle\frac{1}{(u-a_{1})\cdots(u-a_{n})}=\sum_{k=0}^{\infty}h_{k-n}(a_{1},\dots,a_{n})u^{-k},$
$\displaystyle
u^{n}=\sum_{k=0}^{\infty}h_{n-k}(a_{1},\dots,a_{k+1})(u-a_{1})\cdots(u-a_{k}),$
$\displaystyle\frac{1}{u^{n}}=\sum_{k=0}^{\infty}(-1)^{n-k}e_{k-n}(a_{1},\dots,a_{k-1})\frac{1}{(u-a_{1})\cdots(u-a_{k})}.$

Double application of Lemma 7.2 implies the following useful observation.

######  Lemma 7.3.

For any sequence $a=(0,a_{1},a_{2},\dots)$

$\displaystyle\sum_{m=0}^{\infty}\frac{(x|a)^{m}}{(u|\tau
a)^{m}}=\sum_{m=0}^{\infty}\frac{x^{m}}{u^{m}}.$

###### Proof.

$\displaystyle\sum_{m=0}^{\infty}\frac{(x|a)^{m}}{(u|\tau a)^{m}}$
$\displaystyle=\sum_{m,k=0}^{\infty}(-1)^{m-k}e_{m-k}(0,a_{1}\cdots
a_{m-1})x^{k}\frac{1}{(u|\tau a)^{m}}$
$\displaystyle=\sum_{k=0}^{\infty}x^{k}\sum_{m=0}^{\infty}(-1)^{k-m}e_{m-k}(a_{1}\cdots
a_{m-1})\frac{1}{(u|\tau a)^{m}}=\sum_{k=0}^{\infty}\frac{x^{k}}{u^{k}}.$ ∎

Consider a part of the generating function (2.3) of ordinary Schur
$Q$-functions that corresponds only to positive values of $\lambda_{i}$:

$\displaystyle
Q^{+}(u_{1},\dots,u_{l})=\sum_{\lambda_{1},\dots,\lambda_{l}\in{\mathbb{Z}}_{>0}}\frac{Q_{\lambda}}{u_{1}^{\lambda_{1}}\cdots
u_{l}^{\lambda_{l}}}.$

By (7.2), every non-zero coefficient of $Q^{+}(u_{1},\dots,u_{l})$ up to a
sign coincides with a classical Schur $Q$-function labeled by an appropriate
strict partition. In [11] the following remarkable observation is made.

######  Theorem 7.4 ([11]).

For any sequence $a=(0,a_{1},a_{2},\dots)$

$\displaystyle
Q^{+}(u_{1},\dots,u_{l})=\sum_{\lambda_{1},\dots,\lambda_{l}\in{\mathbb{Z}}_{>0}}\frac{Q^{(a)}_{\lambda}}{(u_{1}|\tau
a)^{\lambda_{1}}\cdots(u_{l}|\tau a)^{\lambda_{l}}}.$

###### Proof.

In [11] theorem is proved by induction on the length of the vector $\lambda$.
A very short proof of this theorem follows from Lemma 7.3 and definition
(7.1). Indeed,

$\displaystyle\sum_{\lambda_{i}\in{\mathbb{Z}}_{>0}}\frac{Q^{(a)}_{\lambda}}{(u_{1}|\tau
a)^{\lambda_{1}}\cdots(u_{l}|\tau
a)^{\lambda_{l}}}=\frac{2^{l}}{(N-l)!}\sum_{\sigma\in
S_{N}}\prod_{i=1}^{l}\sum_{\lambda_{i}\in{\mathbb{Z}}_{>0}}\frac{(x_{\sigma(i)}|a)^{\lambda_{i}}}{(u_{i}|\tau
a)^{\lambda_{i}}}\prod_{i\leq l,i<j\leq
N}\frac{x_{\sigma(i)}+x_{\sigma(j)}}{x_{\sigma(i)}-x_{\sigma(j)}}$
$\displaystyle\qquad{}=\frac{2^{l}}{(N-l)!}\sum_{\sigma\in
S_{N}}\prod_{i=1}^{l}\sum_{\lambda_{i}\in{\mathbb{Z}}_{>0}}\frac{x_{\sigma(i)}^{\lambda_{i}}}{u_{i}^{\lambda_{i}}}\prod_{i\leq
l,i<j\leq
N}\frac{x_{\sigma(i)}+x_{\sigma(j)}}{x_{\sigma(i)}-x_{\sigma(j)}}=Q^{+}(u_{1},\dots,u_{l}).$
∎

Thus, Theorem 7.4 suggests that multiparameter Schur $Q$-functions are
obtained from classical Schur $Q$-functions by the change of the basis of
expansion $\big{\\{}1/u^{k}\big{\\}}\to\big{\\{}1/(u|\tau a)^{k}\big{\\}}$ in
the generating function $Q^{+}(u_{1},\dots,u_{l})$.

Lemma 7.2 immediately implies the following relations between classical and
multiparameter Schur $Q$-functions (see also [8, Theorem 10.2])

######  Corollary 7.5.

For any sequence of complex numbers $a=(0,a_{1},a_{2},\dots)$ and any integer
vector $\alpha=(\alpha_{1},\dots,\alpha_{l})$ with
$\alpha_{i}\in{\mathbb{Z}}_{>0}$,

$\displaystyle
Q^{(a)}_{\alpha}=\sum_{\lambda_{1},\dots,\lambda_{l}\in{\mathbb{Z}}_{>0}}(-1)^{\sum\lambda_{i}-\sum\alpha_{i}}e_{\alpha_{1}-\lambda_{1}}(a_{1},\dots,a_{\alpha_{1}-1})\cdots
e_{\alpha_{l}-\lambda_{l}}(a_{1},\dots,a_{\alpha_{l}-1}){Q_{\lambda}}.$

######  Theorem 7.6.

For any sequence of complex numbers $a=(0,a_{1},a_{2},\dots)$ and any integer
vector $\alpha=(\alpha_{1},\dots,\alpha_{l})$ with
$\alpha_{i}\in{\mathbb{Z}}_{>0}$, multiparameter Schur $Q$-function
$Q^{(a)}_{\alpha}$ is a solution of (5.1).

###### Proof.

The constant polynomial $1$ is obviously a solution of (5.1) in
$\mathcal{B}_{\rm odd}$. By the vertex operator presentation (3.4) and
Corollary 7.5,

$\displaystyle
Q^{(a)}_{\alpha}=\sum_{\lambda_{1},\dots,\lambda_{l}\in{\mathbb{Z}}_{>0}}A_{\lambda_{1},\alpha_{1}}\cdots
A_{\lambda_{l},\alpha_{l}}\varphi_{\lambda_{l}}\cdots\varphi_{\lambda_{1}}\cdot
1$

with $A_{n,k}=(-1)^{n-k}e_{k-n}(a_{1},\dots,a_{k-1})$. Hence,

$\displaystyle Q^{(a)}_{\alpha}=X_{\alpha_{l}}\cdot X_{\alpha_{1}}\cdot 1,$

where
$X_{m}=\sum\limits_{s>0}(-1)^{m-s}e_{s-m}(a_{1},\dots,a_{s-1})\varphi_{s}$,
and Corollary 6.2 proves the statement. ∎

### Acknowledgements

The author would like to thank the referee for the thoughtful and careful
review that helped to improve the text of the paper.

## References

  * [1] Alldridge A., Sahi S., Salmasian H., Schur $Q$-functions and the Capelli eigenvalue problem for the Lie superalgebra $\mathfrak{q}(n)$, in Representation Theory and Harmonic Analysis on Symmetric Spaces, Contemp. Math., Vol. 714, Amer. Math. Soc., Providence, RI, 2018, 1–21, arXiv:1701.03401.
  * [2] Date E., Jimbo M., Kashiwara M., Miwa T., Transformation groups for soliton equations. IV. A new hierarchy of soliton equations of KP-type, Phys. D 4 (1982), 343–365.
  * [3] Date E., Kashiwara M., Jimbo M., Miwa T., Transformation groups for soliton equations, in Nonlinear integrable Systems – Classical Theory and Quantum Theory (Kyoto, 1981), World Sci. Publishing, Singapore, 1983, 39–119.
  * [4] Date E., Kashiwara M., Miwa T., Transformation groups for soliton equations. II. Vertex operators and $\tau$ functions, Proc. Japan Acad. Ser. A Math. Sci. 57 (1981), 387–392.
  * [5] Ikeda T., Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian, Adv. Math. 215 (2007), 1–23, arXiv:math.AG/0508110.
  * [6] Ikeda T., Mihalcea L.C., Naruse H., Factorial $P$\- and $Q$-Schur functions represent equivariant quantum Schubert classes, Osaka J. Math. 53 (2016), 591–619, arXiv:1402.0892.
  * [7] Ikeda T., Naruse H., Excited Young diagrams and equivariant Schubert calculus, Trans. Amer. Math. Soc. 361 (2009), 5193–5221, arXiv:math.AG/0703637.
  * [8] Ivanov V.N., Interpolation analogues of Schur $Q$-functions, J. Math. Sci. 131 (2005), 5495–5507, arXiv:math.CO/0305419.
  * [9] Jarvis P.D., Yung C.M., Symmetric functions and the KP and BKP hierarchies, J. Phys. A: Math. Gen. 26 (1993), 5905–5922.
  * [10] Jimbo M., Miwa T., Solitons and infinite-dimensional Lie algebras, Publ. Res. Inst. Math. Sci. 19 (1983), 943–1001.
  * [11] Korotkikh S., Dual multiparameter Schur $Q$-functions, J. Math. Sci. 224 (2017), 263–268.
  * [12] Kvinge H., Ozan Oğuz C., Reeks M., The center of the twisted Heisenberg category, factorial $P$-Schur functions, and transition functions on the Schur graph, Sém. Lothar. Combin. 80B (2018), Art. 76, 12 pages, arXiv:1712.09626.
  * [13] Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1995.
  * [14] Miwa T., Jimbo M., Date E., Solitons: differential equations, symmetries and infinite-dimensional algebras, Cambridge Tracts in Mathematics, Vol. 135, Cambridge University Press, Cambridge, 2000.
  * [15] Nazarov M., Capelli identities for Lie superalgebras, Ann. Sci. École Norm. Sup. (4) 30 (1997), 847–872, arXiv:q-alg/9610032.
  * [16] Sahi S., Salmasian H., Serganova V., The Capelli eigenvalue problem for Lie superalgebras, arXiv:1807.07340.
  * [17] Sato M., Soliton equations as dynamical systems on infinite-dimensional Grassmann manifold, in Nonlinear Partial Differential Equations in Applied Science (Tokyo, 1982), North-Holland Math. Stud., Vol. 81, North-Holland, Amsterdam, 1983, 259–271.
  * [18] You Y., Polynomial solutions of the BKP hierarchy and projective representations of symmetric groups, in Infinite-Dimensional Lie Algebras and Groups (Luminy-Marseille, 1988), Adv. Ser. Math. Phys., Vol. 7, World Sci. Publ., Teaneck, NJ, 1989, 449–464.
  * [19] You Y., DKP and MDKP hierarchy of soliton equations, Phys. D 50 (1991), 429–462.