Super Major Index Cyclic Sieving

TL;DR

This paper proves a cyclic sieving phenomenon for signed standard tableaux of rectangular shape using a super major index, extending known results to more general shapes via Cartesian products.

Contribution

It introduces a CSP for signed tableaux with a super major index, generalizing Rhoades's CSP to non-rectangular shapes through Cartesian products.

Findings

CSP holds for signed tableaux of rectangular shape.

The super major index generates the sieving polynomial.

Results extend to arbitrary shapes via Cartesian products.

Abstract

Recently, Armon and Swanson introduced signed standard tableaux and a corresponding super major index that refines the classical major index. In this paper, we prove that signed standard tableaux of rectangular shape exhibit a cyclic sieving phenomenon (CSP) under the combined action of Sch\"utzenberger promotion and cyclic shift of the signs, with the sieving polynomial given by the super major index generating function. This extends Rhoades's celebrated CSP for standard Young tableaux. Furthermore, by considering Cartesian products of tableaux, we generalize this result to arbitrary non-rectangular shapes.