A diagram-like basis for the multiset partition algebra

TL;DR

This paper introduces a diagram-like basis for the multiset partition algebra, facilitating the understanding of its structure and representations through visual diagrammatic methods, analogous to the classical partition algebra.

Contribution

It constructs a new diagram-like basis for the multiset partition algebra and uses it to develop irreducible representations and a change-of-basis, enhancing structural understanding.

Findings

Constructed a diagram-like basis for the multiset partition algebra.

Derived irreducible representations using the new basis.

Established a change-of-basis similar to that of the partition algebra.

Abstract

There is a classical connection between the representation theory of the symmetric group and the general linear group called Schur-Weyl duality. Variations on this principle yield analogous connections between the symmetric group and other objects such as the partition algebra and more recently the multiset partition algebra. The partition algebra has a well-known basis indexed by graph-theoretic diagrams which allows the multiplication in the algebra to be understood visually as combinations of these diagrams. We construct an analogous basis for the multiset partition algebra called the diagram-like basis and use this basis to construct its irreducible representations and give a generating set. We also provide a change-of-basis from the orbit basis of the multiset partition algebra to this diagram-like basis which exhibits similarities to the analogous change of basis for the partition algebra.