Super Major Index Cyclic Sieving

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.