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Action of $W$-type operators on Schur functions and Schur Q-functions

Xiaobo Liu, Chenglang Yang

TL;DR

This work develops a uniform vertex-operator framework for a family of $W$-type differential operators $\{\widetilde{P}^{(k)}_m,\widehat{P}^{(k)}_m\}$ acting on Schur functions $S_\lambda$ and Schur $Q$-functions $Q_\lambda$. It proves explicit, uniform action formulas for these operators on both families of symmetric functions, linking $W$-constraints in higher KdV hierarchies to Virasoro and cut-and-join structures. The paper then applies these formulas to derive two cornerstone results: Alexandrov’s conjecture for the Brézin-Gross-Witten tau-function and Mironov-Morozov’s formula for the Kontsevich-Witten tau-function, both expressed as simple $Q$-function expansions. The approach leverages Hall-Littlewood functions and vertex operators to provide a streamlined, extensible method that could extend to broader Hall-Littlewood families and illuminate W-constraints in integrable systems and matrix models.

Abstract

In this paper, we investigate a series of W-type differential operators, which appear naturally in the symmetry algebras of KP and BKP hierarchies. In particular, they include all operators in the W-constraints for tau functions of higher KdV hierarchies which satisfy the string equation. We will give simple uniform formulas for actions of these operators on all ordinary Schur functions and Schur's Q-functions. As applications of such formulas, we will give new simple proofs for Alexandrov's conjecture and Mironov-Morozov's formula, which express the Brézin-Gross-Witten and Kontsevich-Witten tau-functions as linear combinations of Q-functions with simple coefficients respectively.

Action of $W$-type operators on Schur functions and Schur Q-functions

TL;DR

This work develops a uniform vertex-operator framework for a family of -type differential operators acting on Schur functions and Schur -functions . It proves explicit, uniform action formulas for these operators on both families of symmetric functions, linking -constraints in higher KdV hierarchies to Virasoro and cut-and-join structures. The paper then applies these formulas to derive two cornerstone results: Alexandrov’s conjecture for the Brézin-Gross-Witten tau-function and Mironov-Morozov’s formula for the Kontsevich-Witten tau-function, both expressed as simple -function expansions. The approach leverages Hall-Littlewood functions and vertex operators to provide a streamlined, extensible method that could extend to broader Hall-Littlewood families and illuminate W-constraints in integrable systems and matrix models.

Abstract

In this paper, we investigate a series of W-type differential operators, which appear naturally in the symmetry algebras of KP and BKP hierarchies. In particular, they include all operators in the W-constraints for tau functions of higher KdV hierarchies which satisfy the string equation. We will give simple uniform formulas for actions of these operators on all ordinary Schur functions and Schur's Q-functions. As applications of such formulas, we will give new simple proofs for Alexandrov's conjecture and Mironov-Morozov's formula, which express the Brézin-Gross-Witten and Kontsevich-Witten tau-functions as linear combinations of Q-functions with simple coefficients respectively.
Paper Structure (12 sections, 16 theorems, 153 equations)

This paper contains 12 sections, 16 theorems, 153 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representing $P^{(k)}_m$ in terms of vertex operators
  4. Actions of W-type operators $\widehat{P}^{(k)}_m$ on Q-functions
  5. Proof of Theorem \ref{['thm:main2']}
  6. Special cases of Theorem \ref{['thm:main2']}
  7. Application to BGW model
  8. Application to Kontsevich-Witten model
  9. Actions of W-type operators $\widetilde{P}^{(k)}_m$ on Schur functions
  10. Proof of Theorem \ref{['thm:main1']}
  11. Special cases of Theorem \ref{['thm:main1']}
  12. Conflict of interest and data availability statement

Key Result

Theorem 1.1

When $\rho=-1$, we set $\widehat{P}^{(k)}_m=P^{(k)}_m(-1)$. For any $k\in\mathbb{Z}_+, m\in\mathbb{Z}$, and $\lambda=(\lambda_1,\dots,\lambda_l)\in\mathbb{Z}^l$, the action of $\widehat{P}^{(k)}_m$ on $Q_\lambda$ is given by where $\lambda-m\epsilon_i :=(\lambda_1,\dots,\lambda_i-m,\dots,\lambda_l)$, and for any integer $n$,

Theorems & Definitions (27)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Remark 3.1
  • Lemma 3.2
  • Theorem 3.3
  • Remark 3.4
  • Corollary 3.5
  • Lemma 4.1
  • Lemma 4.2
  • ...and 17 more