Factoring the Dedekind-Frobenius determinant of a semigroup

TL;DR

This paper extends the factorization of the Dedekind-Frobenius determinant from groups to semigroups, providing explicit factorizations for various classes and applications to Frobenius semigroup algebras and matrix algebra structures.

Contribution

It introduces explicit factorizations of the semigroup determinant for commutative and inverse semigroups, generalizing classical results and applying to Frobenius algebra structures.

Findings

Explicit factorization for commutative semigroups

Recovery of Wilf-Lindström and Wood factorizations

Frobenius property of semigroup algebras over finite rings

Abstract

The representation theory of finite groups began with Frobenius's factorization of Dedekind's group determinant. In this paper, we consider the case of the semigroup determinant. The semigroup determinant is nonzero if and only if the complex semigroup algebra is Frobenius, and so our results include applications to the study of Frobenius semigroup algebras. We explicitly factor the semigroup determinant for commutative semigroups and inverse semigroups. We recover the Wilf-Lindstr\"om factorization of the semigroup determinant of a meet semilattice and Wood's factorization for a finite commutative chain ring. The former was motivated by combinatorics and the latter by coding theory over finite rings. We prove that the algebra of the multiplicative semigroup of a finite Frobenius ring is Frobenius over any field whose characteristic doesn't divide that of the ring. As a consequence we obtain an easier proof of Kov\'acs's theorem that the algebra of the monoid of matrices over a finite field is a direct product of matrix algebras over group algebras of general linear groups (outside of the characteristic of the finite field).