A counterexample to a conjecture on Cartan determinants of monoid algebras

TL;DR

This paper provides a counterexample to a recent conjecture by Steinberg, demonstrating that the Cartan matrix of a monoid algebra over the complex numbers can be non-singular while being singular over a field of positive characteristic.

Contribution

The authors construct a specific finite monoid that disproves Steinberg's conjecture about Cartan matrices in monoid algebras.

Findings

Counterexample to Steinberg's conjecture established

Cartan matrix properties differ between characteristic zero and positive characteristic

Disproves the universality of the conjecture for all finite monoids

Abstract

We show that there are finite monoids $M$ such that the Cartan matrix of the monoid algebra $\mathbb C M$ is non-singular, whilst the Cartan matrix of $kM$ is singular for some field $k$ of positive characteristic, disproving a recent conjecture of Steinberg.