Descent sets for symplectic groups

TL;DR

This paper introduces a descent set for oscillating tableaux that parallels the role of descent sets in standard tableaux, providing combinatorial insights into symplectic group representations and their branching rules.

Contribution

It defines a new descent set for oscillating tableaux and demonstrates its preservation under Sundaram's correspondence, linking combinatorics to symplectic representation theory.

Findings

Descent set for oscillating tableaux is introduced.

Descent set is preserved by Sundaram's correspondence.

Provides combinatorial interpretation of symplectic group branching rules.

Abstract

The descent set of an oscillating (or up-down) tableau is introduced. This descent set plays the same role in the representation theory of the symplectic groups as the descent set of a standard tableau plays in the representation theory of the general linear groups. In particular, we show that the descent set is preserved by Sundaram's correspondence. This gives a direct combinatorial interpretation of the branching rules for the defining representations of the symplectic groups; equivalently, for the Frobenius character of the action of a symmetric group on an isotypic subspace in a tensor power of the defining representation of a symplectic group.