On the spherical partition algebra

Abstract

For $k\in \mathbb{N}$ we introduce an idempotent subalgebra, the spherical partition algebra ${\mathcal{S}\mathcal{P}}_k$, of the partition algebra ${\mathcal{P}}_k$, that we define using an embedding associated with the trivial representation of the symmetric group ${\mathfrak{S}}_k$. We determine a basis for ${\mathcal{S}\mathcal{P}}_k$ and this provides a combinatorial interpretation of the dimension of ${\mathcal{S}\mathcal{P}}_k$, involving bipartite partitions of $k$. For $t\in \mathbb{C}$ we consider the specialized algebra ${\mathcal{S}\mathcal{P}}_k(t)$. For $t=n\in \mathbb{N}$, we describe the structure of ${\mathcal{S}\mathcal{P}}_k(n)$ by giving the permutation module decomposition of the $k$'th symmetric power of the defining module for the symmetric group algebra $\mathbb{C}{\mathfrak{S}}_n$. In general, we show that ${\mathcal{S}\mathcal{P}}_k(t)$ is quasi-hereditary over $\mathbb{C}$ for all $t\in \mathcal{C}$, except $t=0$. We determine the decomposition numbers for ${\mathcal{S}\mathcal{P}}_k(t)$ for every specialization $t\in \mathbb{C}$ except $t=0$, (which includes semisimple and non-semisimple cases). In particular we determine the structure of all indecomposable projective modules, and the indecomposable tilting modules.

27 pages, final version accepted for publication in Israel Journal of Mathematics