Applications of representation theory of finite semigroups

Jorge Almeida (University of Porto)

Monday 26th January, 2009 15:00-16:00 Mathematics Building, room 203


The representation theory of semigroups started in the early 1940's with the work of A. H. Clifford. But it was in the mid-1950's that it received a major boost with the work W. D. Munn, starting from his very first published paper, and also independently by I. S. Ponizovsky. In particular, they characterized those finite semigroups whose semigroup algebra over a given field is semisimple. In the late 1960's, J. Rhodes further investigated the specific case of finite semigroups, with the aim of computing the so-called group complexity of finite semigroups, a programme which succeeded in the case of semigroups in which every element lies in some subgroup and which remains an open problem in the general case. In recent work, we have extended the ideas of J. Rhodes from representations over the complex field to arbitrary fields, which leads to various applications, from representation theory to the theories of formal languages and automata: which finite semigroups admit faithful triangular representations; unambiguous products of rational languages; Cerny's conjecture for some classes of automata.

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