Partition algebras ${\mathsf{P}}_k(n)$ with $2k>n$ and the fundamental theorems of invariant theory for the symmetric group ${\mathsf{S}}_n$

Abstract

Assume ${\mathsf{M}}_n$ is the $n$-dimensional permutation module for the symmetric group ${\mathsf{S}}_n$, and let ${\mathsf{M}}_n^{\otimes k}$ be its $k$-fold tensor power. The partition algebra ${\mathsf{P}}_k(n)$ maps surjectively onto the centralizer algebra ${\mathsf{End}}_{{\mathsf{S}}_n}({\mathsf{M}}_n^{\otimes k})$ for all $k,n\in {\mathbb{Z}}_{\ge 1}$ and isomorphically when $n\ge 2k$. We describe the image of the surjection ${\Phi }_{k,n}:{\mathsf{P}}_k(n)\to {\mathsf{End}}_{{\mathsf{S}}_n}({\mathsf{M}}_n^{\otimes k})$ explicitly in terms of the orbit basis of ${\mathsf{P}}_k(n)$ and show that when $2k>n$ the kernel of ${\Phi }_{k,n}$ is generated by a single essential idempotent ${\mathsf{e}}_{k,n}$, which is an orbit basis element. We obtain a presentation for ${\mathsf{End}}_{{\mathsf{S}}_n}({\mathsf{M}}_n^{\otimes k})$ by imposing one additional relation, ${\mathsf{e}}_{k,n}=0$, to the standard presentation of the partition algebra ${\mathsf{P}}_k(n)$ when $2k>n$. As a consequence, we obtain the fundamental theorems of invariant theory for the symmetric group ${\mathsf{S}}_n$. We show under the natural embedding of the partition algebra ${\mathsf{P}}_n(n)$ into ${\mathsf{P}}_k(n)$ for $k\ge n$ that the essential idempotent ${\mathsf{e}}_{n,n}$ generates the kernel of ${\Phi }_{k,n}$. Therefore, the relation ${\mathsf{e}}_{n,n}=0$ can replace ${\mathsf{e}}_{k,n}=0$ when $k\ge n$.

Cited by in corpus (1)

• Polynomial invariants on matrices and partition, Brauer algebra