Algebra of double cosets of a symmetric group by a smaller symmetric group

TL;DR

This paper studies the algebraic structure of double cosets in symmetric groups, providing explicit formulas, generators, relations, and an interpolating family of algebras depending on a complex parameter.

Contribution

It introduces explicit multiplication formulas, describes the algebra via generators and relations, and constructs a new interpolating family of algebras depending on a complex parameter.

Findings

Explicit formulas for multiplication in the algebra

Description of the algebra via generators and relations

Construction of an interpolating family of algebras

Abstract

Fix a natural $\alpha$. Let $n\ge \alpha$ be an integer. Consider the symmetric group $S_{\alpha+n}$ and its subgroup $S_n$. We consider the group algebra of $S_{\alpha+n}$ and its subalgebra $\mathbb{O}[\alpha;n]$ consisting of $S_n$-biinvariant functions, i.e., functions, which are constant on double cosets of $S_{\alpha+n}$ with respect to $S_n$. We obtain two simple descriptions of $\mathbb{O}[\alpha;n]$. First, we write explicitly formulas for multiplication in a natural basis (structure constants are Pochhammer symbols). Secondly, we describe this algebra in terms of generators and relations. We also construct an interpolating family of algebras $\mathbb{O}[\alpha;\nu]$ depending on a complex parameter $\nu$.