The Algebra of Conjugacy Classes in Symmetric Groups and Partial Permutations

TL;DR

This paper proves a convolution formula for conjugacy classes in symmetric groups, introduces a new semigroup of partial permutations, and explores its structure, representations, and related algebraic properties.

Contribution

It provides a proof of a conjectured convolution formula and introduces a novel semigroup of partial permutations with detailed structural analysis.

Findings

Convolution formula for conjugacy classes proved

New semigroup of partial permutations characterized

Filtrations on invariant subalgebra discussed

Abstract

We prove a convolution formula for the conjugacy classes in symmetric groups conjectured by the second author. A combinatorial interpretation of coefficients is provided. As a main tool we introduce new semigroup of partial permutations. We describe its structure, representations, and characters. We also discuss filtrations on the subalgebra of invariants in the semigroup algebra.