Conjugacy Classes of Renner Monoids

TL;DR

This paper classifies conjugacy classes in Renner monoids, linking them to Weyl group actions, and establishes their relation to irreducible representations, generalizing known concepts from rook monoids.

Contribution

It provides a comprehensive description of conjugacy classes in Renner monoids, introduces a formula for their count, and connects conjugacy to representation theory, extending Munn conjugacy to this context.

Findings

Conjugacy classes correspond to Weyl group orbits.

A formula for counting conjugacy classes is derived.

Munn conjugacy coincides with other conjugacy notions in Renner monoids.

Abstract

In this paper we describe conjugacy classes of a Renner monoid $R$ with unit group $W$, the Weyl group. We show that every element in $R$ is conjugate to an element $ue$ where $u\in W$ and $e$ is an idempotent in a cross section lattice. Denote by $W(e)$ and $W_*(e)$ the centralizer and stabilizer of $e\in \Lambda$ in $W$, respectively. Let $W(e)$ act by conjugation on the set of left cosets of $W_*(e)$ in $W$. We find that $ue$ and $ve$ ($u, v\in W$) are conjugate if and only if $uW_*(e)$ and $vW_*(e)$ are in the same orbit. As consequences, there is a one-to-one correspondence between the conjugacy classes of $R$ and the orbits of this action. We then obtain a formula for calculating the number of conjugacy classes of $R$, and describe in detail the conjugacy classes of the Renner monoid of some $\cal J$-irreducible monoids. We then generalize the Munn conjugacy on a rook monoid to any Renner monoid and show that the Munn conjugacy coincides with the semigroup conjugacy, action conjugacy, and character conjugacy. We also show that the number of inequivalent irreducible representations of $R$ over an algebraically closed field of characteristic zero equals the number of the Munn conjugacy classes in $R$.