Combinatorial Gelfand Models for Semisimple Diagram Algebras

Mazorchuk, Volodymyr

doi:10.1007/s00032-013-0206-2

Cited by 4 publications

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“…Analogous results have been obtained for many other important families of semigroups, such as full linear monoids [26,29,38], endomorphism monoids of various algebraic structures [15,18,30,31,37], sandwich semigroups [13,14] and, more recently, certain families of diagram monoids [22,24,25,58].The current article mainly concerns the partial Brauer monoid PB n and the Motzkin monoid M n (see Section 2 for definitions). There is a growing body of literature on partial Brauer monoids and algebras; 1 see for example [7,16,43,46,55,60,[64][65][66]. Motzkin monoids and algebras are a more recent phenomenon [7,9,17,46,66,70], and are planar versions of the partial Brauer monoids and algebras (see Section 2); the relationship between PB n and M n is therefore analogous to that between the Brauer monoid B n and the Jones monoid J n .…”

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“…There is a growing body of literature on partial Brauer monoids and algebras; 1 see for example [7,16,43,46,55,60,[64][65][66]. Motzkin monoids and algebras are a more recent phenomenon [7,9,17,46,66,70], and are planar versions of the partial Brauer monoids and algebras (see Section 2); the relationship between PB n and M n is therefore analogous to that between the Brauer monoid B n and the Jones monoid J n . 2 The planarity built into the definition of the Motzkin and Jones monoids leads to an aperiodic structure (all subgroups are trivial), and can have interesting effects when comparing the complexity of certain problems; for example, the enumeration of idempotents is far more difficult in M n and J n than in PB n and B n [16,17], while it is quite the opposite for the enumeration of minimal-size idempotent generating sets of the singular ideals of J n and B n [25].…”

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confidence: 99%

“…The current article mainly concerns the partial Brauer monoid PB n and the Motzkin monoid M n (see Section 2 for definitions). There is a growing body of literature on partial Brauer monoids and algebras; 1 see for example [7,16,43,46,55,60,[64][65][66]. Motzkin monoids and algebras are a more recent phenomenon [7,9,17,46,66,70], and are planar versions of the partial Brauer monoids and algebras (see Section 2); the relationship between PB n and M n is therefore analogous to that between the Brauer monoid B n and the Jones monoid J n .…”

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Motzkin monoids and partial Brauer monoids

2017

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We study the partial Brauer monoid and its planar submonoid, the Motzkin monoid. We conduct a thorough investigation of the structure of both monoids, providing information on normal forms, Green's relations, regularity, ideals, idempotent generation, minimal (idempotent) generating sets, and so on. We obtain necessary and sufficient conditions under which the ideals of these monoids are idempotent-generated. We find formulae for the rank (smallest size of a generating set) of each ideal, and for the idempotent rank (smallest size of an idempotent generating set) of the idempotent-generated subsemigroup of each ideal; in particular, when an ideal is idempotent-generated, the rank and idempotent rank are equal. Along the way, we obtain a number of results of independent interest, and we demonstrate the utility of the semigroup theoretic approach by applying our results to obtain new proofs of some important representation theoretic results concerning the corresponding diagram algebras, the partial (or rook) Brauer algebra and Motzkin algebra

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Motzkin monoids and partial Brauer monoids

2017

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“…A Gelfand model of a finite-dimensional semisimple algebra A is a multiplicity-free additive generator of A-mod; see [1,13,27] for further details.…”

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“…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].…”

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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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Gelfand models for diagram algebras

Halverson

Reeks

2014

J Algebr Comb

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A Gelfand model for a semisimple algebra A over an algebraically closed field K is a linear representation that contains each irreducible representation of A with multiplicity exactly one. We give a method of constructing these models that works uniformly for a large class of semisimple, combinatorial diagram algebras including the partition, Brauer, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, and planar rook monoid algebras. In each case, the model representation is given by diagrams acting via "signed conjugation" on the linear span of their horizontally symmetric diagrams. This representation is a generalization of the Saxl model for the symmetric group. Our method is to use the Jones basic construction to lift the Saxl model from the symmetric group to each diagram algebra. In the case of the planar diagram algebras, our construction exactly produces the irreducible representations of the algebra.

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Combinatorial Gelfand Models for Semisimple Diagram Algebras

Cited by 4 publications

References 29 publications

Motzkin monoids and partial Brauer monoids

Motzkin monoids and partial Brauer monoids

Jucys–Murphy elements and Grothendieck groups for generalized rook monoids

Gelfand models for diagram algebras

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