The modular representation theory of monoids and a conjecture on the Cartan determinant of a monoid algebra

Abstract

This paper proposes the conjecture that if the Cartan matrix of the complex algebra of a finite monoid is nonsingular, then the Cartan matrix of the algebra of that monoid over any field is nonsingular. The conjecture is proved for von Neumann regular monoids and for monoids whose left (or right) stabilizers are aperiodic. Both these classes contain the class of finite groups, for which the conjecture holds by classical results of Brauer. The basics of modular representation theory for finite monoids is developed in order to elaborate on this conjecture and to prove some of the partial results.