Symmetry of the generating function of semistandard oscillating tableaux
Masato Kobayashi, Tomoo Matsumura, Shogo Sugimoto
TL;DR
The article extends the theory of Schur and quasi-symmetric function expansions to semistandard oscillating tableaux (SSOT), proving that the SSOT generating function is $F$-positive and, in its homogeneous components, symmetric with Schur-positivity. Using a type C combinatorial calculus, the authors generalize Gessel's theorem and Assaf-Searles' refinements to SSOTs, and establish a Sullivan-style Burge–Sundaram correspondence that yields a product formula for the SSOT generating function in terms of $s_ u$. They further derive explicit Schur expansions via vertical-strip constructions and show Saturated Newton Polytope (SNP) properties and partial orthogonality, refining the understanding of the Newton polytope for these generating functions. The work connects SSOTs to King tableaux and Burge arrays, providing a coherent combinatorial framework that mirrors classic symmetric-function theory while illuminating applications to SNP and polyhedral geometry. Overall, the paper advances symmetry, positivity, and polyhedral aspects of SSOTs, enriching the algebraic combinatorics of oscillating tableaux and their connections to Lusztig weights and crystal theory.
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.