Symmetry of the generating function of semistandard oscillating tableaux

TL;DR

This paper investigates the generating function of semistandard oscillating tableaux (SSOT), demonstrating its symmetry, Schur-positivity, and $F$-positivity, thus advancing understanding of their algebraic and combinatorial properties.

Contribution

It extends existing expansions to SSOTs, proves the symmetry and Schur-positivity of their generating function, and shows it has a saturated Newton polytope.

Findings

The generating function of SSOTs is $F$-positive.

The generating function is symmetric and Schur-positive.

The generating function has a saturated Newton polytope.

Abstract

H.Choi-D.Kim-S.J.Lee and S.J.Lee introduced a new kind of tableaux, semistandard oscillating tableaux (SSOT), around 2024 in the context of Lusztig $q$-weight multiplicities, KR crystals and King tableaux. In this paper, we study generating function of the SSOTs and its symmetry. First, we extend Gessel's and Assaf-Searles' expansion of a Schur function in terms of fundamental quasi-symmetric functions to our generating function. As a consequence, we show that it is $F$-positive. Further, we improve Sundaram's work on oscillating tableaux by proving that it is symmetric, Schur-positive, and has Saturated Newton polytope.