Simplicity, primitivity and semiprimitivity of etale groupoid algebras with applications to inverse semigroup algebras

This paper studies simplicity, primitivity and semiprimitivity of algebras associated to \'etale groupoids. Applications to inverse semigroup algebras are presented. The results also recover the semiprimitivity of Leavitt path algebras and can be used to recover the known primitivity criterion for Leavitt path algebras.Comment: Updated after referee report and corrected misprint

A Groupoid Approach to Discrete Inverse Semigroup Algebras

Let $K$ be a commutative ring with unit and $S$ an inverse semigroup. We show that the semigroup algebra $KS$ can be described as a convolution algebra of functions on the universal \'etale groupoid associated to $S$ by Paterson. This result is a simultaneous generalization of the author's earlier work on finite inverse semigroups and Paterson's theorem for the universal $C^*$-algebra. It provides a convenient topological framework for understanding the structure of $KS$, including the center and when it has a unit. In this theory, the role of Gelfand duality is replaced by Stone duality. Using this approach we are able to construct the finite dimensional irreducible representations of an inverse semigroup over an arbitrary field as induced representations from associated groups, generalizing the well-studied case of an inverse semigroup with finitely many idempotents. More generally, we describe the irreducible representations of an inverse semigroup $S$ that can be induced from associated groups as precisely those satisfying a certain "finiteness condition". This "finiteness condition" is satisfied, for instance, by all representations of an inverse semigroup whose image contains a primitive idempotent

The averaging trick and the Cerny conjecture

The results of several papers concerning the \v{C}ern\'y conjecture are deduced as consequences of a simple idea that I call the averaging trick. This idea is implicitly used in the literature, but no attempt was made to formalize the proof scheme axiomatically. Instead, authors axiomatized classes of automata to which it applies

M\"obius Functions and Semigroup Representation Theory II: Character formulas and multiplicities

We generalize the character formulas for multiplicities of irreducible constituents from group theory to semigroup theory using Rota's theory of M\"obius inversion. The technique works for a large class of semigroups including: inverse semigroups, semigroups with commuting idempotents, idempotent semigroups and semigroups with basic algebras. Using these tools we are able to give a complete description of the spectra of random walks on finite semigroups admitting a faithful representation by upper triangular matrices over the complex numbers. These include the random walks on chambers of hyperplane arrangements studied by Bidigare, Hanlon, Rockmere, Brown and Diaconis. Applications are also given to decomposing tensor powers and exterior products of rook matrix representations of inverse semigroups, generalizing and simplifying earlier results of Solomon for the rook monoid.Comment: Some minor typos corrected and references update