A domino tableau-based view on type B Schur-positivity

TL;DR

This paper extends the concept of Schur-positivity to type B signed permutations using domino tableaux and Chow's quasisymmetric functions, establishing new bijections and demonstrating their Schur-positivity.

Contribution

It introduces a novel type B Schur-positivity framework for signed permutations based on domino tableaux and compatible with Coxeter group descent algebra.

Findings

Signed arc permutations are type B Schur-positive.

Descent-preserving bijections link signed permutations to domino tableaux.

The framework aligns with Solomon's descent algebra for Coxeter groups.

Abstract

Over the past years, major attention has been drawn to the question of identifying Schur-positive sets, i.e. sets of permutations whose associated quasisymmetric function is symmetric and can be written as a non-negative sum of Schur symmetric functions. The set of arc permutations, i.e. the set of permutations $\pi$ in $S_n$ such that for any $1\leq j \leq n$, $\{\pi(1),\pi(2),\dots,\pi(j)\}$ is an interval in $\mathbb{Z}_n$ is one of the most noticeable examples. This paper introduces a new type B extension of Schur-positivity to signed permutations based on Chow's quasisymmetric functions and generating functions for domino tableaux. As an important characteristic, our development is compatible with the works of Solomon regarding the descent algebra of Coxeter groups. In particular, we design descent preserving bijections between signed arc permutations and sets of domino tableaux to show that they are indeed type B Schur-positive.