Skew quantum Murnaghan-Nakayama rule

TL;DR

This paper generalizes the skew Murnaghan-Nakayama rule to a broader identity involving skew Schur functions and quantum power sums, providing two proofs and proposing conjectures for Hall-Littlewood polynomials.

Contribution

It introduces a new, more general identity for skew Schur functions and quantum power sums, with two distinct proofs and conjectures for further extensions.

Findings

Derived a new expansion formula for skew Schur functions and quantum power sums.

Provided bijective and algebraic proofs of the main identity.

Proposed conjectures for skew rules of Hall-Littlewood polynomials.

Abstract

In this paper, we extend recent results of Assaf and McNamara on skew Pieri rule and skew Murnaghan-Nakayama rule to a more general identity, which gives an elegant expansion of the product of a skew Schur function with a quantum power sum function in terms of skew Schur functions. We give two proofs, one completely bijective in the spirit of Assaf-McNamara's original proof, and one via Lam-Lauve-Sotille's skew Littlewood-Richardson rule. We end with some conjectures for skew rules for Hall-Littlewood polynomials.