On a conjecture on 2-reduced Schur functions and Schur's Q-functions

TL;DR

This paper proves a conjecture relating sums of products of 2-reduced Schur functions and Schur's Q-functions, using new expressions and combinatorial methods, advancing understanding in algebraic combinatorics.

Contribution

It introduces a new expression for Schur's Q-functions and proves the conjecture in previously unproven cases, expanding the theoretical framework.

Findings

Proof of the conjecture in new cases

Introduction of a new expression for Schur's Q-functions

Use of inverse Kostka matrix combinatorics

Abstract

Motivated by Sato and Mori's work on the Korteweg-de Vries (KdV) equation and the modified KdV equation, Mizukawa, Nakajima, and Yamada made a conjecture on 2-reduced Schur functions and Schur's Q-functions. The conjecture claims that certain sums of products of a Littlewood-Richardson coefficient and two 2-reduced Schur functions are equal to Schur's Q-functions up to a scalar multiple. In this paper we give a proof of the conjecture in cases which have not been proved yet. We introduce a new expression of Schur's Q-functions and use it to prove the conjecture. Combinatorics of the inverse Kostka matrix is also used. We also provide consideration of the conjecture in general case.