Action of $W$-type operators on Schur functions and Schur 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 Brézin-Gross-Witten and Kontsevich-Witten tau-functions as linear combinations of Q-functions with simple coefficients respectively.

32 pages, accepted for publication in Journal of the London Mathematical Society

References in corpus (15)

• Tautological relations and the r-spin Witten conjecture
• An algebro-geometric proof of Witten's conjecture
• Polynomial averages in the Kontsevich model
• A new cohomology class on the moduli space of curves
• Cut-and-Join operator representation for Kontsevich-Witten tau-function
• Cut-and-join description of generalized Brezin-Gross-Witten model
• Superintegrability of Kontsevich matrix model
• Pfaffian Formulas and Schur Q-Function Identities
• CFT approach to constraint operators for ($\beta$-deformed) hermitian one-matrix models
• Generalized Q-functions for GKM
• Generalized Brézin-Gross-Witten tau-function as a hypergeometric solution of the BKP hierarchy
• The Green polynomials via vertex operators
• Virasoro versus superintegrability. Gaussian Hermitian model
• Action of Virasoro operators on Hall-Littlewood polynomials
• Virasoro Action on the $Q$-Functions