Boson-fermion correspondence for Hall-Littlewood polynomials revisited

TL;DR

This paper explores the relationship between Hall-Littlewood polynomials, vertex operators, and fermionic actions, establishing a boson-fermion correspondence and linking symmetric functions to KP hierarchy tau-functions.

Contribution

It introduces a new boson-fermion correspondence for Hall-Littlewood polynomials and connects deformed symmetric functions to integrable systems.

Findings

Established a boson-fermion correspondence for Hall-Littlewood polynomials.

Linked elements orthogonal to Schur functions with KP hierarchy tau-functions.

Connected twisted vertex operators with classical Heisenberg algebra.

Abstract

We connect twisted vertex operator presentation of Hall-Littlewood polynomials with the action of charged free fermions, describe a boson-fermion correspondence that relates twisted vertex operators with classical Heisenberg algebra. We also show that the elements of the orthogonal to Schur functions basis of the one-parameter deformation of the ring of symmetric functions $\Lambda[t]$ are $\tau$-functions of KP hierarchy.

Equations

$$s_{\lambda }(x_1,x_2,\dots )=\frac{\det [x_i^{{\lambda }_j+n-j}]}{\det [x_i^{n-j}]}$$

$$h_r(x_1,x_2\dots )=\sum _{1\le i_1\le \cdots \le i_r<\infty }{x}_{i_1}\dots x_{i_r},$$

$$e_r(x_1,x_2\dots )=\sum _{1<i_1<\cdots <i_r<\infty }{x}_{i_1}\dots x_{i_r},$$

$$p_k(x_1,x_2,\dots )=\sum _i{x}_i^k.$$

$$\Lambda =\mathbb{C}[h_1,h_2,\dots ]=\mathbb{C}[e_1,e_2,\dots ]=\mathbb{C}[p_1,p_2,\dots ].$$

$$\mathcal{B}={\oplus }_{m\in \mathbb{Z}}{\mathcal{B}}^{(m)},{\mathcal{B}}^{(m)}=z^m\cdot \mathbb{C}[p_1,p_2,\dots ]=z^m\Lambda .$$

$$H(u)=\sum _{k\ge 0}\frac{h_k}{u^k},E(u)=\sum _{k\ge 0}\frac{e_k}{u^k},P(u)=\sum _{k=1}^{\infty }{p}_k{u}^k.$$

$$H(u)=\prod _{i\ge 1}\frac{1}{1-x_i/u},E(u)=\prod _{i\ge 1}(1+x_i/u),$$

$$H(u)E(-u)=1,$$

$$H(u)=exp\left(\sum _{n\ge 1}\frac{p_n}{n}\frac{1}{u^n}\right),E(u)=exp\left(-\sum _{n\ge 1}\frac{(-1{)}^n{p}_n}{n}\frac{1}{u^n}\right).$$

$$[{\alpha }_k,{\alpha }_n]=k{\delta }_{k,-n}\cdot 1.$$

$$[\alpha (u),\alpha (v)]={\partial }_v\delta (u,v),$$

$$\{\begin{array}{ll}{\alpha }_{-n}=p_n/n,& n>0,\\ {\alpha }_n=\frac{\partial }{\partial p_n},& n>0,\\ {\alpha }_0=m.& \end{array}$$

$$⟨s_{\lambda },s_{\mu }⟩={\delta }_{\lambda ,\mu }.$$

$$E^{\perp }(u)=\sum _{k\ge 0}{e}_k^{\perp }{u}^k,H^{\perp }(u)=\sum _{k\ge 0}{h}_k^{\perp }{u}^k.$$

$$E^{\perp }(u)=exp\left(-\sum _{k\ge 1}(-1{)}^k\frac{\partial }{\partial p_k}{u}^k\right),H^{\perp }(u)=exp\left(\sum _{k\ge 1}\frac{\partial }{\partial p_k}{u}^k\right).$$

$$\left(1-\frac{u}{v}\right)E^{\perp }(u)E(v)=E(v)E^{\perp }(u),$$

$$\left(1-\frac{u}{v}\right)H^{\perp }(u)H(v)=H(v)H^{\perp }(u),$$

$$H^{\perp }(u)E(v)=\left(1+\frac{u}{v}\right)E(v)H^{\perp }(u),$$

$$E^{\perp }(u)H(v)=\left(1+\frac{u}{v}\right)H(v)E^{\perp }(u).$$

$$R(u)(z^{m}f)={\left(\frac{z}{u}\right)}^{m+1}f.$$

$$R^{-1}(u)(z^{m}f)={\left(z\right)}^{m-1}{u}^{m}f.$$

$${\Phi }^{+}(u)$$

$${\Phi }^{-}(u)$$

$${\Phi }^{+}(u){|}_{{\mathcal{B}}^{(m)}}$$

$${\Phi }^{-}(u){|}_{{\mathcal{B}}^{(m)}}$$

$${\Phi }^{\pm }(u){\Phi }^{\pm }(v)+{\Phi }^{\pm }(v){\Phi }^{\pm }(u)$$

$${\Phi }^{+}(u){\Phi }^{-}(v)+{\Phi }^{-}(v){\Phi }^{+}(u)$$

$${\Phi }^{+}(u){\Phi }^{+}(v)(z^{m}f)$$

$${\Phi }^{-}(u){\Phi }^{-}(v)(z^{m}f)$$

$$\left(1-\frac{u}{v}\right){\Phi }^{+}(u){\Phi }^{-}(v)(z^{m}f)=z^m{u}^{-m}{v}^{m-1}H(u)E(-v)E^{\perp }(-u)H^{\perp }(v)(f),$$

$$\left(1-\frac{v}{u}\right){\Phi }^{-}(v){\Phi }^{+}(u)(z^{m}f)=z^m{u}^{-m-1}{v}^{m}H(u)E(-v)E^{\perp }(-u)H^{\perp }(v)(f).$$

$${\left(1-\frac{u}{v}\right)}^{-1}\sum _{k\ge 0}\frac{u^k}{v^{k+1}},{\left(1-\frac{v}{u}\right)}^{-1}=\sum _{k\ge 0}\frac{v^k}{u^{k+1}}.$$

Full text

1. Introduction

Vertex operator realization of Hall-Littlewood polynomials through generalized fermions was first described in [9], followed by the construction of a deformed version of boson-fermion correspondence in [10]. The ideas were further developed and applied in works of many authors, such as [1], [2], [4], [5], [15].

The first goal of this note is to show that generalized fermions in [9] can be interpreted as a simple twist of charged free fermions that provide vertex operator realization of Schur symmetric functions described in [11], [14], [17].

The deformed boson-fermion correspondence of [10] establishes connection between the actions of generalized fermions and the twisted Heisenberg algebra. This correspondence found its applications in [5], [15]. Our second goal is to propose a version of boson-fermion correspondence that relates generalized fermions with the action of the classical Heisenberg algebra. This construction is different from the deformed boson-fermion correspondence in [10] and has its own advantages: it does not require adjustments in the standard definition of the normal ordered product of fields; the action of classical Heisenberg algebra and, as a consequence, of the Virasoro algebra are naturally present in the picture; the action of the twisted Heisenberg algebra appears as a renormalization of the action of the classical Heisenberg algebra; proofs of this version of the boson-fermion correspondence, the original deformed boson-fermion correspondence of [10], and its applications become simple implications of a twist of the classical boson-fermion correspondence [3].

The third result of this paper is the construction of tau-functions of the KP hierarchy in the deformed ring of symmetric functions $\Lambda[t]$. In [14] symmetric functions $S_{\lambda}$ are introduced as a basis dual to the basis of classical Schur functions $s_{\lambda}$ with respect to the natural scalar form in the deformed ring of symmetric functions $\Lambda[t]$. Extending the idea of twisting the fields of charged free fermions we prove that $S_{\lambda}$’s are solutions of the bilinear KP identity and provide two different vertex operator realizations of this family of symmetric functions. Note that this is stronger than the result of [5] that proves that the element $\Lambda[t]$ of the form $\sum s_{\lambda}S_{\lambda}$ is a tau-function of the KP hierarchy.

In Sections 2, 3 we review necessary facts about symmetric functions and the action of charged free fermions on the ring of symmetric functions. In Section 4 we connect twisted fermions with classical charged free fermions. In Section 5 we propose a version of boson-fermion correspondence for twisted fermions and compare it with the construction of [10]. In Section 6 we prove that symmetric functions $S_{\lambda}$ that form a basis of $\Lambda[t]$ orthogonal to the basis of Schur functions $s_{\lambda}$ are tau-functions of the KP hierarchy. We provide two different versions of their vertex operator realization.

2. Preliminaries on symmetric functions

We review necessary facts about symmetric functions following [14], [16]. Let $\Lambda=\Lambda[x_{1},x_{2},\dots]$ be the ring of symmetric functions in variables $(x_{1},x_{2},\dots)$. Schur symmetric functions $s_{\lambda}$ labeled by partitions $\lambda=(\lambda_{1}\geq\lambda_{2}\geq\dots\geq\lambda_{n}\geq 0)$ defined by

[TABLE]

form a linear basis of $\Lambda$. Complete symmetric functions $h_{r}=s_{(r)}$ are given by

[TABLE]

elementary symmetric functions $e_{r}=s_{(1^{r})}$ by

[TABLE]

and power sums $p_{k}$ by

[TABLE]

It will be convenient to set $h_{-k}(x_{1},x_{2}\dots)=e_{-k}(x_{1},x_{2}\dots)=p_{-k}(x_{1},x_{2}\dots)=0$ for $k>0$ and $h_{0}=e_{0}=p_{0}=1$. Each of these families generate the ring of symmetric functions $\Lambda$ is a polynomial ring:

[TABLE]

Based on the interpretation of a polynomial ring as the ring of symmetric functions, one defines boson Fock space $\mathcal{B}$. Let $\mathcal{B}={\mathbb{C}}[z,z^{-1},p_{1},p_{2},\dots]$ be the graded space of polynomials

[TABLE]

We write generating functions for complete, elementary symmetric functions and power sums:

[TABLE]

Then

[TABLE]

[TABLE]

and

[TABLE]

Heisenberg algebra is a complex Lie algebra generated by elements $\{\alpha_{m}\}_{m\in{\mathbb{Z}}}$ and central element $1$ with commutation relations

[TABLE]

Combining generators into formal distribution $\alpha(u)=\sum_{k}\alpha_{k}{u^{-k-1}}$ we can rewrite this relation as

[TABLE]

where $\delta(u,v)=\sum_{k\in{\mathbb{Z}}}{u^{k}}{v^{-k-1}}$ is the formal delta distribution. There is a natural action of Heisenberg algebra on the graded components $\alpha_{n}:\mathcal{B}^{(m)}\to\mathcal{B}^{(m)}$ defined by multiplication and differentiation operators:

[TABLE]

The ring of symmetric functions $\Lambda$ possesses a natural scalar product, where classical Schur functions $s_{\lambda}$ constitute an orthonormal basis:

[TABLE]

Then for the operator of multiplication by a symmetric function $f$ one can define the adjoint operator $f^{\perp}$ acting on the ring of symmetric functions by the standard rule: $\langle f^{\perp}g,w\rangle=\langle g,fw\rangle$, where $g,f,w\in\Lambda$. We consider generating functions of the adjoint operators

[TABLE]

One can prove that

[TABLE]

The following commutation relations serve as the foundation of most of calculations in this note.

Proposition 2.1**.**

[14]**

[TABLE]

Remark 2.1*.*

Statements of Proposition 2.1 should be understood as equalities of series expansions in powers of $u^{k}v^{-m}$ for $k,m\geq 0$.

3. Fermions and Schur symmetric functions

Following [3], [6], [12], define the action of algebra of charged free fermions on the boson Fock space.

Let $R(u):\mathcal{B}\to\mathcal{B}$ act on elements $z^{m}f$, $f\in\Lambda$ by the rule

[TABLE]

(see e.g.[12], [13]). Then $R^{-1}(u)$ acts as

[TABLE]

One should think of $R^{\pm 1}(u)$ as operators that transport the action of other operators along the grading of the boson Fock space: $R^{\pm 1}(u):\mathcal{B}^{(m)}\to\mathcal{B}^{(m\pm 1)}$. We set

[TABLE]

Observe that

[TABLE]

Proposition 3.1**.**

Quantum fields $\Phi^{\pm}(u)$ satisfy the relations of the algebra of charged free fermions:

[TABLE]

Here $\delta(u,v)=\sum_{k\in{\mathbb{Z}}}\frac{u^{k}}{v^{k+1}}$ is formal delta distribution.

Proof.

We use Proposition 2.1 to prove this classical result, thus illustrating the simplicity of this approach. Relations between other vertex operators further in this text are proved along the same lines.

Using commutation relations of Proposition 2.1, for any $f\in\Lambda$,

[TABLE]

[TABLE]

Changing the roles of $u$ and $v$ in these calculations one gets (3.3).

For (3.4) observe that

[TABLE]

[TABLE]

Note that by Remark 2.1,

[TABLE]

Then

[TABLE]

We used (2.3) along with the property of formal delta distribution $\delta(u,v)A(v)=\delta(u,v)A(u)$ for any formal distribution $A(u)$ (see e.g. [12], [13]). ∎

Let $1\in\mathcal{B}^{(0)}$ be the constant function.

Proposition 3.2**.**

[TABLE]

where

[TABLE]

Proof.

Using Proposition 2.1 and that $E^{\perp}(-u)(1)=1$, we write:

[TABLE]

∎

It is known [14], [11] that $Q(u_{1},\dots,u_{l})$ is the generating function for Schur symmetric functions in the following sense. Consider the series expansion of the rational function

[TABLE]

in the region $|u_{1}|<\dots<|u_{l}|$. Then for any partition $\lambda=(\lambda_{1}\geq\dots\geq\lambda_{l}\geq 0)$ the coefficient of $u_{1}^{-\lambda_{1}}\dots u_{l}^{-\lambda_{l}}$ is exactly Schur symmetric function: $Q_{(\lambda_{1},\dots,\lambda_{l})}=s_{\lambda}$. Thus, Proposition 3.2 describes vertex operator presentation of Schur symmetric functions (cf. [11], [14], [17]).

4. Generalized fermions and Hall-Littlewood symmetric functions.

Let $\lambda$ be a partition of length at most $n$, let $t$ be a parameter. Hall-Littlewood polynomials are defined by

[TABLE]

where $m(i)$ is the number of parts of the partition $\lambda$ that are equal to $i$, and $S_{n}$ is the symmetric group of $n$ letters [14]. Labeled by partitions Hall-Littlewood polynomials form a linear basis of the deformed ring $\Lambda[t]$ of symmetric polynomials with coefficients in ${\mathbb{C}}[t]$.

In this section we show that vertex operators presentation of Hall-Littlewood polynomials $P_{\lambda}$ is obtained from vertex operators presentation of Schur symmetric functions $s_{\lambda}$ by a simple twist of the fields of charged free fermions by multiplication by $E(-u/t)$ or $H(u/t)$. This approach significantly simplifies the technical proofs of [9], [10] and provides new insight into the original results of these papers.

Consider the deformed boson Fock space $\mathcal{B}(t)=\oplus\mathcal{B}^{(m)}[t]$, where $\mathcal{B}^{(m)}[t]=z^{m}\Lambda[t]$. We extend the action of the operators in Section 2 to $\mathcal{B}(t)$ by $t$-linearity. Define quantum fields of operators acting on on $\mathcal{B}(t)$

[TABLE]

Proposition 4.1**.**

Quantum fields $\Psi^{\pm}(u)$ satisfy relations of generalized fermions

[TABLE]

Proof.

The proof is based on commutation relations of Proposition 2.1 and follows the same lines as the proof of Proposition 3.1. ∎

Proposition 4.1 immediately implies that operators $\Psi^{\pm}(u)$ provide vertex operators realization [9] of Hall-Littlewood polynomials. Let

[TABLE]

where the expression $\prod_{1\leq i<j\leq l}\frac{u_{j}-u_{i}}{u_{j}-u_{i}t}$ is understood as the series expansion of this rational function in the region $|u_{1}|<\dots<|u_{l}|$. Consider the expansion

[TABLE]

in the region $|u_{1}|<\dots<|u_{l}|$. It is proved in [9] (see also [14]) that for any partition $\lambda=(\lambda_{1}\geq\dots\geq\lambda_{l}\geq 0)$ the coefficient of $u_{1}^{-\lambda_{1}}\dots u_{l}^{-\lambda_{l}}$ is exactly Hall-Littlewood symmetric function: $F_{\lambda}=P_{\lambda}(x_{1},x_{2},\dots;t)$.

Proposition 4.2**.**

(Vertex operator presentation of Hall-Littlewood symmetric functions) One has

[TABLE]

Proof.

From definition (4.1), (4.2) and commutation relations of Proposition 2.1 one immediately finds that

[TABLE]

which simplifies to $z^{l}u_{1}^{-l}\dots u_{l}^{-1}F(u_{1},\dots,u_{l};t)$ since $E^{\perp}(-u)(1)=1$. ∎

Corollary 4.1**.**

Generating functions $F(u_{1},\dots,u_{l};t)$ for Hall-Littlewood polynomials and $Q(u_{1},\dots,u_{l})$ for Schur symmetric functions are related by the formula

[TABLE]

5. Boson-fermion correspondence for Hall-Litllewood polynomials revisited

The classical boson-fermion correspondence connects the action of charged free fermions and the action of (classical) Heisenberg algebra with generators $\{\alpha_{m}\}_{m\in{\mathbb{Z}}}$, the central element $1$, and relations (2.5):

(I)  Heisenberg algebra $\to$ charged free fermions $\to$ Heisenberg algebra.

In the same spirit a deformed boson-fermion correspondence between the actions of generalized fermions and twisted Heisenberg algebra was established in [10]:

(II)  twisted Heisenberg algebra $\to$ generalized fermions $\to$ twisted Heisenberg algebra.

The twisted Heisenberg algebra is defined as an algebra with generators $\{\mathbb{h}_{k}\}$, central element $c$, and relations

[TABLE]

This construction found its applications in [5], [15]. At the same time, it has certain disadvantageous deviations from the format of the classical boson-fermion correspondence. In particular, to obtain the bosonisation [10] of generalized fermions one has to change the standard definition of normal ordered product of fields. Moreover, the natural presence of the action of the classical Heisenberg algebra and the Virasoro algebra is not reflected by this deformed version.

In this section we propose another deformed construction of boson-fermion correspondence, different from [10]. It establishes connection between the actions of generalized fermions and classical Heisenberg algebra:

(III)  Heisenberg algebra $\to$ generalized fermions $\to$ Heisenberg algebra.

Among the advantages of the boson-fermion correspondence (III) over (II) is that (a) the standard definition of the normal ordered product of fields is used in all definitions; (b) the action of the twisted Heisenberg algebra (5.1) is present as certain renormalization of the action of the classical Heisenberg algebra; (c) the action of Virasoro algebra remains naturally in the construction; (d) proofs of statements of correspondences (II) and (III) become simple implications of the results of classical boson-fermion correspondence (I).

We recall the statement of the classical boson-fermion correspondence in the form convenient for our exposition. For a formal distribution $a(z)=\sum_{n\in{\mathbb{Z}}}a_{n}z^{-n-1}$ define $a(z)_{+}=\sum_{n\leq-1}a_{n}z^{-n-1},\quad a(z)_{-}=\sum_{n\geq 0}a_{n}z^{-n-1}.$ The normal ordered product of two formal distributions is the formal distribution defined by the formula $:a(z)b(z):=a(z)_{+}b(z)+b(z)a_{(}z)_{-}$ (see e.g. [13] ).

Proposition 5.1**.**

(Classical Boson-fermion correspondence (I), [3],[6], [12], )

1. a)

Consider the action (2.7) of the Heisenberg algebra on the boson space $\mathcal{B}$. Then the fields

[TABLE]

satisfy relations (3.3), (3.4) and define the action of charged free fermions on $\mathcal{B}$. 2. b)

Let $\Phi^{\pm}(u)=\sum_{k\in{\mathbb{Z}}}\Phi^{\pm}_{k+{1}/2}u^{\pm k}$ satisfy relations (3.3), (3.4). Introduce

[TABLE]

Then coefficients of the formal distribution

[TABLE]

satisfy the relations (2.5) of Heisenberg algebra. 3. c)

Let $\Phi^{\pm}(u)=\sum_{k\in{\mathbb{Z}}}\Phi^{\pm}_{k+{1}/2}u^{\pm k}$ satisfy relations (3.3), (3.4). For any $\beta\in{\mathbb{C}}$, the coefficients of the formal distribution $L^{(\beta)}(u)=\sum_{k\in{\mathbb{Z}}}L_{k}u^{-k-2}$ defined by the formula

[TABLE]

satisfy the relations of Virasoro algebra with central charge $c_{\beta}=-12\beta^{2}+12\beta-2$:

[TABLE]

Proof.

Given the action (2.7) of the Heisenberg algebra, we use presentations (2.4), (2.9) to compare (5.2) and (5.3) with the definition (3.1) and (3.2) of $\Phi^{\pm}(u)$ to get part a), while for the proofs of b) and c) we refer to [13]. ∎

Proposition 5.2**.**

(Revisited version Boson-fermion correspondence (III))

1. a)

The action of the twisted Heisenberg algebra can be defined by renormalization of the action of the classical Heisenberg algebra:

[TABLE] 2. b)

Consider the action (2.7) of Heisenberg algebra on the boson space $\mathcal{B}$ expanded to the deformed space $\mathcal{B}(t)$ by the rule $\alpha_{n}(t^{s}f)=t^{s}\alpha_{n}(f)$. Then the fields

[TABLE]

satisfy relations (4.3), (4.4) and define the action of generalized fermions on $B(t)$. 3. c)

The other way, consider the action of generalized fermions $\Psi^{\pm}(u)$ on $B(t)$ defined by (4.1), (4.2). Then the coefficients of the normal ordered product

[TABLE]

satisfy relations (2.5) and define the action of the classical Heisenberg algebra on $\mathcal{B}(t)$. 4. d)

Given the action (4.1), (4.2) of generalized fermions $\Psi^{\pm}(u)$ on $\mathcal{B}(t)$, the coefficients of the formal distribution $L^{(\beta)}(u)=\sum_{k\in{\mathbb{Z}}}L_{k}u^{-k-2}$ defined by the formula

[TABLE]

satisfy the relations of the Virasoro algebra with central charge $c_{\beta}=-12\beta^{2}+12\beta-2$.

Proof.

Direct check of commutation relations proves part a). Given the action (2.7) of the Heisenberg algebra we use presentations (2.4), (2.9) to compare (5.5), (5.6) with definition (4.1), (4.2) of $\Psi^{\pm}(u)$ to obtain part b). Note that from part a) and (4.1), (4.2) quantum fields

[TABLE]

satisfy relations of charged free fermions. Then part c) follows from Proposition 5.1 part b).

Part d) follows from the substitution of (5.7) into the formula of $L^{(\beta)}(u)$ in Proposition 5.1 part c) and the property [14]

[TABLE]

∎

6. Tau-functions of the KP hierarchy in $\Lambda[t]$

The ring $\Lambda[t]$ possesses a scalar product $\langle\cdot,\cdot\rangle_{t}$, which is a deformation of the scalar product (2.8) on $\Lambda$. Following III.4 in [14], define symmetric functions $S_{\lambda}=S_{\lambda}(x_{1},x_{2},\dots;t)$ as a basis dual to classical Schur functions $s_{\lambda}=s_{\lambda}(x_{1},x_{2},\dots)$ with respect to the deformed scalar product:

[TABLE]

Proposition 6.1**.**

Let

[TABLE]

Then for any partition $\lambda$, the coefficient of $u_{1}^{-\lambda_{1}}\dots u_{l}^{-\lambda_{l}}$ is $S_{\lambda}$.

Proof.

By III.4 formula (4.3) and III.2 formula (2.10) in [14], $S_{\lambda}$ can be expressed through Hall-Littlewood polynomials $P_{(k)}$ by an analogue of Jacobi - Trudi formula:

[TABLE]

Here $P_{(k)}=P_{(k)}(x_{1},x_{2},\dots;t)$ is the coefficient of the expansion

[TABLE]

Then

[TABLE]

∎

Corollary 6.1**.**

Generating functions $S(u_{1},\dots,u_{l})$ for symmetric functions $S_{\lambda}$ and $Q(u_{1},\dots,u_{l})$ for Schur symmetric functions $s_{\lambda}$ are related by the formula

[TABLE]

The vertex operator presentation of the generating function $S(u_{1},\dots,u_{l})$ can be written as

[TABLE]

In [5] the relation of symmetric functions $S_{\lambda}$ to tau-functions of the KP hierarchy is discussed. The determinant-type property (6.1) of symmetric functions $S_{\lambda}$ is interpreted as plücker coordinates-type property. This observation allows the authors of [5] to conclude that the expression $\sum_{\lambda}s_{\lambda}S_{\lambda}$ is an example of a tau-function of the KP hierarchy in $\Lambda[t]$.

Here we show that symmetric functions $S_{\lambda}$’s themselves are tau-functions of the KP hierarchy; this result is not present in [5] and does not follow from the mentioned above statement. Moreover, we provide the explicit formula for the charged free fermions action that realize the KP hierarchy for these tau-functions and conclude with one more vertex operator presentation of the generating function of $S_{\lambda}$’s.

As it is well-known, the bilinear form of the KP hierarchy is the equation

[TABLE]

where $\tau\in\mathcal{B}^{(0)}=\Lambda={\mathbb{C}}[p_{1},p_{2},\dots]$ and

[TABLE]

It is known that Schur symmetric functions $s_{\lambda}\in\mathcal{B}^{(0)}$ are solutions of the KP hierarchy [3], [7], [8]. Define formally

[TABLE]

Consder

[TABLE]

and the equation

[TABLE]

where $\tau\in\mathcal{B}^{(0)}(t)$. We summarize the statements in the following proposition.

Proposition 6.2**.**

1. a)

Let $\Phi_{t}^{\pm}(u)$ be defined by (6.3), (6.4) with the expansion in the region $|t|<1$. Operators $\Phi_{t}^{\pm}(u)$ satisfy exactly the same relations as classical charged free fermions $\Phi^{\pm}(u)$ in Proposition 3.1. Thus, operators $\Phi_{t}^{\pm}(u)$ provide the action of charged free fermions on the deformed space $\mathcal{B}(t)$, Consequently, the equation (6.6) is the bilinear identity of the KP hierarchy on functions $\tau\in\mathcal{B}(t)$. 2. b)

Symmetric functions $S_{\lambda}$ are solutions of the bilinear identity (6.6). Consequently, symmetric functions $S_{\lambda}$ are tau-functions of the KP hierarchy. 3. c)

Generating function $S(u_{1},\dots,u_{l})$ of symmetric functions $S_{\lambda}$’s has the second vertex operator presentation:

[TABLE]

Proof.

Schur symmetric functions $s_{\lambda}$ are expressed through complete symmetric functions by Jacobi - Trudi formula $s_{\lambda}=\det[h_{\lambda_{i}-i+j}]$, while complete symmetric functions $h_{k}$ in the determinant can be expressed as polynomial functions of power sums $h_{k}=h_{k}(p_{1},p_{2},\dots)$ through the relation

[TABLE]

At the same time, from (6.1) the dual symmetric functions $S_{\lambda}$ are given by $S_{\lambda}=\det[q_{\lambda_{i}-i+j}]$, where symmetric functions $q_{k}=(1-t)P_{(k)}$ are expressed as functions of power sums $q_{k}=q_{k}(p_{1},p_{2},\dots)$ through the relation

[TABLE]

Thus, $S_{\lambda}$ as a function of power sums $(p_{1},p_{2},\dots)$ can be obtained from $s_{\lambda}$ by the substitution of variables $p_{n}\to(1-t^{n})p_{n}$.

Recall that Schur symmetric functions $s_{\lambda}$ are solutions of the bilinear KP identity (6.2). $\Phi^{\pm}(u)$ in (6.2) in terms of operators $p_{i}$’s and $\partial/\partial p_{i}$’s have the form (5.2), (5.3). Hence $S_{\lambda}$ satisfies the bilinear identity

[TABLE]

where

[TABLE]

and $\Phi^{\pm}_{t}(u)$ is obtained from (5.2), (5.3) by the same substitution $p_{n}\to(1-t^{n})p_{n}$:

[TABLE]

Using the geometric series expansion $1/(1-t^{n})=\sum_{i}(t^{i})^{n}$ in the region $|t|<1$ we can write the second exponential factor as

[TABLE]

Note that

[TABLE]

which gives (6.3) for $\Phi_{t}^{+}(u)$, and similarly (6.4) for $\Phi_{t}^{-}(u)$.

Applying the same substitution $p_{n}\to(1-t^{n})p_{n}$ to the result of Proposition 3.2 we get c).

∎