A Sundaram type bijection for SO(3): vacillating tableaux and pairs of standard Young tableaux and orthogonal Littlewood-Richardson tableaux

TL;DR

This paper introduces a bijection for SO(3) that links vacillating tableaux with pairs of standard Young tableaux and orthogonal Littlewood-Richardson tableaux, enabling a new combinatorial approach to Frobenius characters.

Contribution

It presents a novel bijective method for analyzing the tensor powers of SO(3) representations, connecting vacillating tableaux with classical tableaux and expanding Frobenius characters.

Findings

Established a bijection preserving descent sets

Derived the quasi-symmetric expansion of Frobenius characters

Enhanced combinatorial understanding of SO(3) tensor representations

Abstract

Based on the direct-sum-decomposition of the rth tensor power of the defining representation of the special orthogonal group SO(2k+1) one is interested in a bijective approach for determining the Frobenius characters of the isotypic components. In particular this leads us to a bijection between vacillating tableaux and pairs of standard Young tableaux and orthogonal Littlewood-Richardson tableaux, which we present for SO(3). Moreover we introduce the descent set of a vacillating tableau. As our bijection preserves this descent set, we also obtain the quasi-symmetric expansion of the Frobenius characters.