Algebra and Combinatorics Seminar

Date: October 21, 2022

Time: 3:00PM - 4:00PM

Location: BLOC 302

Speaker: Catherine Yan, TAMU

Title: On the Limiting Vacillating Tableaux for Integer Sequences

Abstract: A fundamental identity in the representation theory of the partition algebra is $n^k = \sum_{\lambda} f^\lambda m_k^\lambda$ for $n \geq 2k$, where $\lambda$ ranges over integer partitions of $n$, $f^\lambda$ is the number of standard Young tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of vacillating tableaux of shape $\lambda$ and length $2k$. Using a combination of RSK insertion and jeu de taquin, Halverson and Lewandowski constructed a bijection $DI_n^k$ that maps each integer sequence to a pair consisting of a standard Young tableau and a vacillating tableau. In this talk we show that for a given integer sequence $i$, when $n$ is sufficiently large, the vacillating tableaux determined by $DI_n^k(i)$ become stable when n goe to infinite; the limit is called the limiting vacillating tableau for $i$. We give a characterization of the set of limiting vacillating tableaux and present explicit formulas that enumerate those vacillating tableaux. This is a joint work with Zhanar Berikkyzy, Pamela Harris, Anna Pun and Chenchen Zhao.