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

TL;DR

This paper studies W-type differential operators related to KP and BKP hierarchies, providing uniform formulas for their action on Schur functions and Q-functions, leading to new proofs of important conjectures.

Contribution

It introduces simple, uniform formulas for W-type operators acting on Schur and Q-functions, enabling new proofs of key conjectures in integrable systems.

Findings

Derived uniform formulas for operator actions on Schur functions

Provided new proofs for Alexandrov's conjecture and Mironov-Morozov's formula

Expressed tau-functions as linear combinations of Q-functions

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 Br\'{e}zin-Gross-Witten and Kontsevich-Witten tau-functions as linear combinations of Q-functions with simple coefficients respectively.