Combinatorial Gelfand models for some semigroups and <i>q</i>-rook monoid algebras

Kudryavtseva, Ganna; Mazorchuk, Volodymyr

doi:10.1017/s0013091507001216

Cited by 5 publications

(5 citation statements)

References 17 publications

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“…(c) The vector space G C is a module over U.sl p .C// ˝C COEB, and moreoverG C Š V .ƒ 0 / ˝C V N ; as U. sl p .C// ˝C COEB -modules:Proof. Claim (a) follows from Theorem 6.3, Theorem 6.4 (a), and formulae(22) and(23). Claim (b) follows from Corollary 6.15.Let us now prove Claim (6.16).…”

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confidence: 79%

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Jucys–Murphy elements and Grothendieck groups for generalized rook monoids

Mazorchuk

Srivastava

2022

J. Comb. Algebra

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We consider a tower of generalized rook monoid algebras over the field C of complex numbers and observe that the Bratteli diagram associated to this tower is a simple graph. We construct simple modules and describe Jucys-Murphy elements for generalized rook monoid algebras.Over an algebraically closed field k of positive characteristic p, utilizing Jucys-Murphy elements of rook monoid algebras, for 0 Ä i Ä p 1 we define the corresponding i -restriction and i -induction functors along with two extra functors. On the direct sum G C of the Grothendieck groups of module categories over rook monoid algebras over k, these functors induce an action of the tensor product of the universal enveloping algebra U.sl p .C// and the monoid algebra COEB of the bicyclic monoid B. Furthermore, we prove that G C is isomorphic to the tensor product of the basic representation of U.sl p .C// and the unique infinite-dimensional simple module over COEB, and also exhibit that G C is a bialgebra. Under some natural restrictions on the characteristic of k, we outline the corresponding result for generalized rook monoids.

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confidence: 79%

“…If we replace, in (1), C by an algebraically closed field k of positive characteristic, then our method gives a modular branching rule as well. We construct a Gelfand model for COEC r o R n in Proposition 3.5 which is a generalization of the case r D 1 as considered in [23], see also [27] and [13].…”

Section: Introductionmentioning

confidence: 99%

Jucys–Murphy elements and Grothendieck groups for generalized rook monoids

Mazorchuk

Srivastava

2022

J. Comb. Algebra

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“…This model (sometimes referred to as the involutive Gelfand model) was generalized to wreath products in [APR2,CF], to inverse semigroups in [KM2] and to general diagram algebras in [HRe], [Maz], see also references in these paper for other generalizations. An alternative approach to Gelfand models for certain classes of groups can be found in [CM].…”

Section: Modules Over Generalized Symmetric Groupsmentioning

confidence: 99%

“…In our setup it is fairly straightforward to combine the above model with the construction used in [KM2], [Maz] to produce a Gelfand model for G (ℓ,d) (significantly simplifying arguments from [APR2]). For each f ∈ G (ℓ,d) denote by I f the set of all involutions in…”

Section: Modules Over Generalized Symmetric Groupsmentioning

confidence: 99%

$$G(\ell ,k,d)$$ G ( ℓ , k , d ) -modules via groupoids

Mazorchuk

Stroppel

2015

J Algebr Comb

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In this note we describe a seemingly new approach to the complex representation theory of the wreath product G ≀ S d where G is a finite abelian group. The approach is motivated by an appropriate version of Schur-Weyl duality. We construct a combinatorially defined groupoid in which all endomorphism algebras are direct products of symmetric groups and prove that the groupoid algebra is isomorphic to the group algebra of G ≀ S d. This directly implies a classification of simple modules. As an application, we get a Gelfand model for G ≀ S d from the classical involutive Gelfand model for the symmetric group. We describe the Schur-Weyl duality which motivates our approach and relate it to various Schur-Weyl dualities in the literature. Finally, we discuss an extension of these methods to all complex reflection groups of type G(ℓ, k, d).

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“…The monoid I n is also known in the literature as the rook monoid because of the alternate characterisation of partial permutations by {0, 1}-matrices with at most one non-zero entry in each row and each column; such matrices are in one-to-one correspondence with placements of non-attacking rooks on an n × n chess board. While the representation theory of the rook monoid is well studied (see for example [22,37,39,55,56,71,72,77,81,82]), there has also been substantial recent interest in the representation theory of wreath products G ≀ I n , where G is a group (definitions are given below). See especially the work of Steinberg [83,84] and Mazorchuk and Srivastava [69].…”

Section: Introductionmentioning

confidence: 99%

Presentations for wreath products involving symmetric inverse monoids and categories

Clark¹,

East²

2022

Preprint

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Wreath products involving symmetric inverse monoids/semigroups/categories arise in many areas of algebra and science, and presentations by generators and relations are crucial tools in such studies. The current paper finds such presentations for M ≀ I n , M ≀ Sing(I n ) and M ≀ I. Here M is an arbitrary monoid, I n is the symmetric inverse monoid, Sing(I n ) its singular ideal, and I is the symmetric inverse category.

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Combinatorial Gelfand models for some semigroups and q-rook monoid algebras

Cited by 5 publications

References 17 publications

Jucys–Murphy elements and Grothendieck groups for generalized rook monoids

Jucys–Murphy elements and Grothendieck groups for generalized rook monoids

$$G(\ell ,k,d)$$ G ( ℓ , k , d ) -modules via groupoids

Presentations for wreath products involving symmetric inverse monoids and categories

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