For $ k \in \mathbb{ N}$ we introduce an idempotent subalgebra, the spherical partition algebra ${ \mathcal{ SP} }_{ k}$, 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 ${ \mathcal{ SP} }_{ k}$ and this provides a combinatorial interpretation of the dimension of $\mathcal{ SP}_{ k}$, involving bipartite partitions of $ k$. For $ t \in \mathbb{ C} $ we consider the specialized algebra $\mathcal{ SP}_{ k}(t)$. For $ t = n \in \mathbb{ N}$, we describe the structure of $\mathcal{ SP}_{ k}(n)$ 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 $\mathcal{ SP}_{ k}(t)$ is quasi-hereditary over $ \mathbb{ C}$ for all $ t \in \mathcal{ C}$, except $ t=0$. We determine the decomposition numbers for $\mathcal{ SP}_{ k}(t)$ 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.
Categories: math.RT