Queer dual equivalence graphs

TL;DR

This paper introduces queer dual equivalence graphs, a new framework for proving Schur P-positivity by generalizing dual equivalence with involutions that include a 'queer' involution, and demonstrates its applications.

Contribution

It defines queer dual equivalence, providing a new axiomatic approach to establish Schur P-positivity and distinguishes it from shifted dual equivalence with novel applications.

Findings

Established queer dual equivalence as a new paradigm for Schur P-positivity

Proved the fundamental quasisymmetric generating function is Schur P-positive under this framework

Applied the concept to the product of Schur P-functions

Abstract

We introduce a new paradigm for proving the Schur $P$-positivity. Generalizing dual equivalence, we give an axiomatic definition for a family of involutions on a set of objects to be a queer dual equivalence, and we prove whenever such a family exists, the fundamental quasisymmetric generating function is Schur $P$-positive. In contrast with shifted dual equivalence, the queer dual equivalence involutions restrict to a dual equivalence when the queer involution is omitted. We highlight the difference between these two generalization with a new application to the product of Schur $P$-functions.