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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.01890 |
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Table of Contents:
- 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.