RSK Insertion for Set Partitions and Diagram Algebras

TL;DR

This paper provides combinatorial proofs for identities in the representation theory of partition algebras, using RSK insertion and jeu de taquin, and extends these methods to various diagram algebras.

Contribution

It introduces a novel combinatorial approach using RSK insertion for identities in partition algebra representations and extends this to subalgebras like Brauer and Temperley-Lieb.

Findings

Proves identities relating set partitions and tableaux counts.

Extends RSK-based insertion techniques to diagram algebras.

Provides combinatorial proofs for algebraic identities.

Abstract

We give combinatorial proofs of two identities from the representation theory of the partition algebra $C A_k(n), n \ge 2k$. The first is $n^k = \sum_\lambda f^\lambda m_k^\lambda$, where the sum is over partitions $\lambda$ of $n$, $f^\lambda$ is the number of standard tableaux of shape $\lambda$, and $m_k^\lambda$ is the number of "vacillating tableaux" of shape $\lambda$ and length $2k$. Our proof uses a combination of Robinson-Schensted-Knuth insertion and jeu de taquin. The second identity is $B(2k) = \sum_\lambda (m_k^\lambda)^2$, where $B(2k)$ is the number of set partitions of $\{1, >..., 2k\}$. We show that this insertion restricts to work for the diagram algebras which appear as subalgebras of the partition algebra: the Brauer, Temperley-Lieb, planar partition, rook monoid, planar rook monoid, and symmetric group algebras.