Complex Representations of Finite Monoids II. Highest Weight Categories and Quivers

Putcha, Mohan S.

doi:10.1006/jabr.1997.7395

Cited by 33 publications

(50 citation statements)

References 9 publications

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“…[2,3]. This paper, like [25,26,19,13,1], aims to reconcile semigroup representation theory with representation theory at large.…”

Section: Introductionmentioning

confidence: 99%

“…The poset of regular J -classes is denoted U (S). It was shown by Putcha [19] that over an algebraically closed field of characteristic zero, the module category of a regular semigroup (one in which all elements are regular) is a highest weight category [7] with weight poset U (S). We need one last fact about finite semigroups in order to state and prove the Clifford-Munn-Ponizovskiȋ theorem.…”

Section: Introductionmentioning

confidence: 99%

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On the irreducible representations of a finite semigroup

Ganyushkin

Mazorchuk

Steinberg

2009

Proc. Amer. Math. Soc.

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Abstract. Work of Clifford, Munn and Ponizovskiȋ parameterized the irreducible representations of a finite semigroup in terms of the irreducible representations of its maximal subgroups. Explicit constructions of the irreducible representations were later obtained independently by Rhodes and Zalcstein and by Lallement and Petrich. All of these approaches make use of Rees's theorem characterizing 0-simple semigroups up to isomorphism. Here we provide a short modern proof of the Clifford-Munn-Ponizovskiȋ result based on a lemma of J. A. Green, which allows us to circumvent the theory of 0-simple semigroups. A novelty of this approach is that it works over any base ring.

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“…[2,3]. This paper, like [25,26,19,13,1], aims to reconcile semigroup representation theory with representation theory at large.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On the irreducible representations of a finite semigroup

Ganyushkin

Mazorchuk

Steinberg

2009

Proc. Amer. Math. Soc.

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“…We take a minimalist approach, stating exactly what we need in order to prove our main result. Details can be found in [9,15,24,30,33]. To fix notation, if S is a monoid, we use Irr(S) to denote the set of equivalence classes of irreducible representations of S. The reader should verify that every irreducible representation of a group is by invertible maps.…”

Section: A Mortality Function For Edsmentioning

confidence: 99%

“…Let C : B ×A → G 0 J be the sandwich matrix for J and denote by ρ⊗C the d|B|×d|A| matrix obtained by applying ρ entrywise to C (where we take ρ(0) to be the d×d zero matrix). The following result can be extracted from [33] and [9,Chapter 5]; see also [30] and [31,Chapter 15] for a summary without proofs or [15,23] for module-theoretic statements and proofs. Now we are ready to prove that f (n) = n(n+1)/2 is a superadditive mortality function for monoids in EDS.…”

Section: A Mortality Function For Edsmentioning

confidence: 99%

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Matrix Mortality and the Černý-Pin Conjecture

Almeida

Steinberg

2009

Developments in Language Theory

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Abstract. In this paper, we establish theČerný-Pin conjecture for automata with the property that their transition monoid cannot recognize the language {a, b} * ab{a, b} * . For the subclass of automata whose transition monoids have the property that each regular J -class is a subsemigroup, we give a tight bound on lengths of reset words for synchronizing automata thereby answering a question of Volkov.

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Invariant Algebra and Cuspidal Representations of Finite Monoids

Putcha

1999

Journal of Algebra

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Motivated by the theories of Hecke algebras and Schur algebras, we consider in this paper the algebra ‫ރ‬ M G of G-invariants of a finite monoid M with unit group G. If M is a regular ''balanced'' monoid, we show that ‫ރ‬ M G is a quasi-hereditary algebra. In such a case, we find the blocks of ‫ރ‬ M G to be the ''sections'' of the blocks of ‫ރ‬ M. We go on to develop a theory of cuspidal representations for balanced monoids. We then apply our results to the full transformation semigroup and the multiplicative monoid of triangular matrices over a finite field. ᮊ 1999 Academic Press w x e associated with the Young diagram as in 5; Section 28 , let J s S eS n n and Ä 4 M s S j J j 0 . Ž . n Ž . Then M is a monoid and all idempotents of J are conjugate, a special property shared by reductive monoids. Because the theory of complex representations of finite groups is so well w x developed, we are motivated in 17, 18 to begin a systematic study of complex representations of finite monoids. Let M be a finite monoid with unit group G. By classical semigroup representation theory, the irreducible characters of M are in 1-1 correspondence with those of the maximal subgroups H of regular J J-classes J of M. The irreducible characters of J the groups H thus serve as ''weights'' for the irreducible characters of M J in much the same way as in the representation theory of algebraic groups or Lie algebras. For an irreducible character of H we constructed in J w x q y 17 , left induced, right induced, and semigroup induced characters , ,˜q yand of G and we showed that is a summand of l . Although the characters q and y of G can be expressed explicitly as sums of somẽ induced group characters, the character is quite mysterious.For an algebra A with group G of automorphisms, the algebra A G of w x G-invariants is a natural and classical object of study, cf. 12 . In this paper we study the algebra ‫ރ‬ M G where M is a finite monoid with unit group G. We show that the invariant algebra ‫ރ‬ M G of the local monoid M is a J J quasi-hereditary algebra if and only if J is G-balanced; i.e., every [ JgU U Ž . Ž . For J g U U M , choose e g E J . Let H denote the H H-class of e , that is J J J w x the unit group of e Me . By semigroup representation theory 3; Chap. 5 , J J the irreducible characters of M are in 1-1 correspondence with the weights Irr H , J g U U. We write this as J Irr M ( " Irr H . 1 Ž . J JgU U Correspondingly for J g U U, ‫ރ‬ J s ‫ރ‬ J , 2 Ž . [ gIrr H J Ž . with each ‫ރ‬ J being an ideal of ‫ރ‬ M and a ‫ރ‬ M, ‫ރ‬ M -bimodule. If 0 J g Irr H , then we denote by the character of G obtained by restricting J to G the irreducible character of M associated with . Let ⑀ be a primitive idempotent of ‫ރ‬ H corresponding to . Then J ⌬ s ‫ރ‬ J и ⑀ , ٌ s ⑀ и ‫ރ‬ J * 3 J ‫ރ‬ M and ‫ރ‬ M are quasi-hereditary algebras. J 0 J Let M be a finite regular monoid J, JЈ g U U, JЈ ) J, g Irr H . Let J J follows. Assume without loss of generality that 1 s e ) e ) иии ) e s J J J 0 1 t e . Let s . Having obtained g char H , i ) 0, let s g J t i...

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Complex Representations of Finite Monoids II. Highest Weight Categories and Quivers

Cited by 33 publications

References 9 publications

On the irreducible representations of a finite semigroup

On the irreducible representations of a finite semigroup

Matrix Mortality and the Černý-Pin Conjecture

Invariant Algebra and Cuspidal Representations of Finite Monoids

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