On the spherical partition algebra

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.