Orders of Finite Reductive Monoids

TL;DR

This paper derives formulas to compute the orders of finite reductive monoids, applies them to specific classes including symplectic monoids, and explores their connections to algebraic invariants like Betti numbers.

Contribution

It introduces four formulas for calculating orders of finite reductive monoids and applies these to various classes, including symplectic monoids, with explicit formulas and connections to algebraic topology.

Findings

Formulas for orders of finite reductive monoids are established.

Explicit order formulas for symplectic monoids are provided.

Connections to H-polynomials and Betti numbers are demonstrated.

Abstract

We show four formulas for calculating the orders of finite reductive monoids with zero. As applications, these formulas are then used to calculate the orders of finite reductive monoids induced from the $F_q$-split $\J$-irreducible monoids $\overline {K^*\rho(G_0)}$ where $G_0$ is a simple algebraic group over the algebraic closure of $F_q$, and $\rho: G_0\to GL(V)$ is the irreducible representation associated with any dominant weight. Finally, we give an explicit formula for the orders of finite symplectic monoids associated with the last fundamental dominant weight of type $C_l$; the connections to $H$-polynomials and Betti numbers are shown.