A Sundaram type bijection for SO(2k+1): vacillating tableaux and pairs conisisting of a standard Young tableau and an orthogonal Littlewood-Richardson tableau

Abstract

We present a bijection between vacillating tableaux and pairs consisting of a standard Young tableau and an orthogonal Littlewood-Richardson tableau for the special orthogonal group SO(2k+1) in odd dimension. This bijection is motivated by the direct-sum-decomposition of the rth tensor power of the defining representation of SO(2k + 1). The question of such a bijection was first asked by Sheila Sundaram in the late nineteen-eighties and remained an open question since then. To formulate it, we present an explicit formulation of Kwons Littlewood-Richardson tableaux and define alternative tableaux. We show that our new alternative tableaux are in bijection with Kwons tableaux by giving an explicit bijection. Those new tableaux help us to reduce the problem to a special case. We then define another bijection, which maps a pair consisting of a standard Young tableau and an alternative tableau to a vacillating tableau with desired properties. For SO(3) we present an alternative formulation of our bijection, which proves further properties, that can only be conjectured for general odd dimension. On the combinatorial side we obtain a bijection between Riordan paths and standard Young tableaux with at most 3 rows, all of even length or all of odd length. Moreover we use a suitably defined descent set for vacillating tableaux to determine the quasi-symmetric expansion of the Frobenius characters of the isotypic components.