Linear Algebraic Monoids

Abstract. In this paper we characterize the congruence associated to the direct sum of all irreducible representations of a finite semigroup over an arbitrary field, generalizing results of Rhodes for the field of complex numbers. Applications are given to obtain many new results, as well as easier proofs of several results in the literature, involving: triangularizability of finite semigroups; which semigroups have (split) basic semigroup algebras, two-sided semidirect product decompositions of finite monoids; unambiguous products of rational languages; products of rational languages with counter; andČerný's conjecture for an important class of automata.

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“…In a finite semigroup (or even in an algebraic semigroup [42,48]), the following stability relations hold [25]:…”

Section: Preliminariesmentioning

confidence: 99%

Representation theory of finite semigroups, semigroup radicals and formal language theory

Almeida

Margolis

Steinberg

et al. 2008

Trans. Amer. Math. Soc.

121

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“…Citations of [1] then slowed for more than a decade, though classes of inverse semigroups important for other reasons have turned out to be factorizable, notably the Renner monoids ( [30] Prop. 11.1; [32], section 8.1) and latterly the re ‡ection monoids of Everitt and Fountain [8].…”

Section: Some Historymentioning

confidence: 99%

Factorizable inverse monoids

FitzGerald

2009

Semigroup Forum

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Factorizable inverse monoids constitute the algebraic theory of those partial symmetries which are restrictions of automorphisms; the formal de…nition is that each element is the product of an idempotent and an invertible. This class of monoids has theoretical signi…cance, and includes concrete instances which are important in various contexts. This survey is organised around the idea of group acts on semilattices and contains a large range of examples. Topics also include methods for construction of factorizable inverse monoids, and aspects of their inner structure, morphisms, and presentations.

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“…It is well-known (see [3]) that any affine algebraic monoid M over K is isomorphic with a closed submonoid of the monoid M n = M n (K) of n × n matrices over K; i.e., M has a faithful matrix representation. One can refer to Putcha [13] and Solomon [16] for basic properties of algebraic monoids. We will always assume M is irreducible as a variety.…”

Section: Representation Theorymentioning

confidence: 99%

Representation Theory of Reductive Normal Algebraic Monoids

Doty¹

1999

Trans. Amer. Math. Soc.

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Abstract. New results in the representation theory of "semisimple" algebraic monoids are obtained, based on Renner's monoid version of Chevalley's big cell. (The semisimple algebraic monoids have been classified by Renner.) The rational representations of such a monoid are the same thing as "polynomial" representations of the associated reductive group of units in the monoid, and this representation category splits into a direct sum of subcategories by "homogeneous" degree. We show that each of these homogeneous subcategories is a highest weight category, in the sense of Cline, Parshall, and Scott, and so equivalent with the module category of a certain finite-dimensional quasihereditary algebra, which we show is a generalized Schur algebra in S. Donkin's sense.

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Linear Algebraic Monoids

Cited by 155 publications

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Representation theory of finite semigroups, semigroup radicals and formal language theory

Representation theory of finite semigroups, semigroup radicals and formal language theory

Factorizable inverse monoids

Representation Theory of Reductive Normal Algebraic Monoids

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