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A Sundaram type bijection for $\mathrm{SO}(2k+1)$: vacillating tableaux and pairs consisting of a standard Young tableau and an orthogonal Littlewood-Richardson tableau
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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 $\mathrm{SO}(2k+1)$. This bijection is motivated by the direct-sum-decomposition of the $r$th tensor power of the defining representation of $\mathrm{SO}(2k+1)$. To formulate it, we use Kwon's orthogonal Littlewood-Richardson tableaux and introduce new alternative tableaux they are in bijection with. 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.
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Flagged Littlewood-Richardson tableaux and branching rule for classical groups
New flagged Littlewood-Richardson tableaux count GL_n to O_n branching multiplicities for arbitrary highest weights, with applications to generalized exponents of types B and D.
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