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

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.