Quasi-partition algebra

TL;DR

The paper introduces the quasi-partition algebra as a new centralizer algebra related to the symmetric group, exploring its structure, basis, dimensions, and representations.

Contribution

It constructs a basis, provides a dimension formula, and describes irreducible representations of the quasi-partition algebra, extending the understanding of related algebraic structures.

Findings

Dimension formula in terms of Bell numbers

Explicit basis construction for $QP_k(n)$

Description of irreducible representations and their indexing

Abstract

We introduce the quasi-partition algebra $QP_k(n)$ as a centralizer algebra of the symmetric group. This algebra is a subalgebra of the partition algebra and inherits many similar combinatorial properties. We construct a basis for $QP_k(n)$, give a formula for its dimension in terms of the Bell numbers, and describe a set of generators for $QP_k(n)$ as a complex algebra. In addition, we give the dimensions and indexing set of its irreducible representations. We also provide the Bratteli diagram for the tower of quasi-partition algebras (constructed by letting $k$ range over the positive integers).