Venue: office XX, XX floor of the “Departamento de Engenharia Mecânica e Industrial (DEMI)” Building

Room: 4.6, ed. VIII

Abstract: The concept of a determinant associated with a finite group originates in the work of Dedekind and Frobenius in the late 19\textsuperscript{th} century. This idea was later extended to finite semigroups, where the determinant of the semigroup algebra captures deep structural information. For commutative semigroups, Steinberg provided a factorization of the semigroup determinant, and more recently, the theory has been developed for broader classes, such as semigroups in the pseudovariety $\ensuremath{\mathsf{ECom}}$, where all idempotents commute.

In this talk, we trace the development of computing determinants for finite semigroups, from its classical origins to more recent results. We focus on semigroups whose structure enables determinant computation via partial orders and canonical decompositions, particularly using the notion of $\ll$-smoothness. These developments highlight both the possibilities and limitations in extending the theory beyond commutative and idempotent-commuting settings.

We also discuss recent progress in identifying broader classes of semigroups where determinant computations remain tractable, even when full $\ll$-transitivity is absent. This opens new directions for understanding the algebraic and combinatorial properties that govern the behavior of semigroup determinants.

Venue: office XX, XX floor of the “Departamento de Engenharia Mecânica e Industrial (DEMI)” Building