Representations of the Renner Monoid

TL;DR

This paper extends the representation theory of rook monoids to Renner monoids associated with reductive monoids, providing explicit irreducible representations and character formulas using geometric and algebraic tools.

Contribution

It generalizes the Munn-Solomon representation theory to all Renner monoids and introduces new formulas for characters based on parabolic subgroups.

Findings

Irreducible representations are classified via the type map and associated polytopes.

An analogue of the Munn-Solomon formula for Renner monoids is established.

Complete irreducible representations are determined by those of certain parabolic subgroups.

Abstract

We describe irreducible representations and character formulas of the Renner monoids for reductive monoids, which generalizes the Munn-Solomon representation theory of rook monoids to any Renner monoids. The type map and polytope associated with reductive monoids play a crucial role in our work. It turns out that the irreducible representations of certain parabolic subgroups of the Weyl groups determine the complete set of irreducible representations of the Renner monoids. An analogue of the Munn-Solomon formula for calculating the character of the Renner monoids, in terms of the characters of the parabolic subgroups, is shown.