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

Judith Braunsteiner

Published 2018-01-11Version 1

Based on the direct-sum-decomposition of the $r^{\text{th}}$ tensor power of the defining representation of the special orthogonal group $\mathrm{SO}(2k+1)$ 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 $\mathrm{SO}(3)$. 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.