# 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

## TL;DR

This paper introduces a bijection connecting vacillating tableaux with pairs of standard Young tableaux and orthogonal Littlewood-Richardson tableaux for the special orthogonal group, enhancing understanding of tensor representations.

## Contribution

It establishes a new bijection involving vacillating tableaux and orthogonal Littlewood-Richardson tableaux, utilizing descent sets to analyze Frobenius characters.

## Key findings

- Bijection between vacillating tableaux and tableau pairs for SO(2k+1)
- New alternative tableaux in bijection with orthogonal Littlewood-Richardson tableaux
- Descent set used to determine quasi-symmetric expansion of Frobenius characters

## 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.