The lattice of submonoids of the uniform block permutations containing the symmetric group

TL;DR

This paper investigates the structure of submonoids within the uniform block permutation monoid that include the symmetric group, revealing a distributive lattice structure and connections to integer partitions and module dimensions.

Contribution

It introduces a new partial order on integer partitions to characterize submonoids containing the symmetric group and analyzes the sizes of J-classes in relation to irreducible modules.

Findings

The lattice of submonoids containing the symmetric group is distributive.

Submonoids correspond to downsets in a new partial order on partitions.

Sizes of J-classes relate to sums of squares of module dimensions.

Abstract

We study the lattice of submonoids of the uniform block permutation monoid containing the symmetric group (which is its group of units). We prove that this lattice is distributive under union and intersection by relating the submonoids containing the symmetric group to downsets in a new partial order on integer partitions. Furthermore, we show that the sizes of the $\mathscr{J}$-classes of the uniform block permutation monoid are sums of squares of dimensions of irreducible modules of the monoid algebra.