Representation theory of the monoid of all partial functions on a set and other Ehresmann semigroups

Abstract: Given a finite semigroup S, we can study its linear representations (for this talk - over the field of complex numbers). Semigroups with natural combinatorial structure are clearly of major interest. An important example of such semigroup is the monoid of all partial functions on an n element set, denoted PT_n. A description of its simple modules by induced left Schützenberger modules was obtained in the fifties by Munn and Ponizovskii as part of a more general work on the representation theory of finite semigroups. Unlike group algebras, semigroup algebras are seldom semisimple and therefore have (none-semisimple) projective modules. We give a description of the indecomposable projective modules of PT_n which is similar in spirit to the Munn-Ponizovskii construction of the simple modules. Moreover, we generalize both results and describe the simple and the indecomposable projective modules of a certain class of Ehresmann semigroups, with the case of PT_n being a natural example. This is a joint work with Stuart Margolis.

operator algebrasrings and algebras

Audience: researchers in the topic

Western Sydney University Abend Seminars

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