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

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.