A Sundaram type Bijection for SO(3): Vacillating Tableaux and Pairs of Standard Young Tableaux and Orthogonal Littlewood-Richardson Tableaux

  • Judith Jagenteufel
Keywords: Special Orthogonal Groups, Vacillating Tableaux, Branching Rules, Riordan Paths


Motivated by the direct-sum-decomposition of the $r^{\text{th}}$ tensor power of the defining representation of the special orthogonal group $\mathrm{SO}(2k + 1)$, we present a bijection between vacillating tableaux and pairs consisting of a standard Young tableau and an orthogonal Littlewood-Richardson tableau for $\mathrm{SO}(3)$.
Our bijection preserves a suitably defined descent set. Using it we determine the quasi-symmetric expansion of the Frobenius characters of the isotypic components.
On the combinatorial side we obtain a bijection between Riordan paths and standard Young tableaux with 3 rows, all of even length or all of odd length.
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