Schur–Weyl dualities for symmetric inverse semigroups

Kudryavtseva, Ganna; Mazorchuk, Volodymyr

doi:10.1016/j.jpaa.2007.12.004

Cited by 11 publications

(10 citation statements)

References 21 publications

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“…These monoids can be thought of as finite analogues of linear algebraic monoids [67,76] and their representation theory gives information on their groups of units, which are groups of Lie type [69,74,75]. On a similar note, applications of semigroup representation theory to Schur-Weyl duality can be found in [46,89].…”

Section: Introductionmentioning

confidence: 99%

Quivers of monoids with basic algebras

Margolis

Steinberg

2012

Compositio Math.

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AbstractWe compute the quiver of any finite monoid that has a basic algebra over an algebraically closed field of characteristic zero. More generally, we reduce the computation of the quiver over a splitting field of a class of monoids that we term rectangular monoids (in the semigroup theory literature the class is known asDO) to representation-theoretic computations for group algebras of maximal subgroups. Hence in good characteristic for the maximal subgroups, this gives an essentially complete computation. Since groups are examples of rectangular monoids, we cannot hope to do better than this. For the subclass of ℛ-trivial monoids, we also provide a semigroup-theoretic description of the projective indecomposable modules and compute the Cartan matrix.

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Section: Introductionmentioning

confidence: 99%

Quivers of monoids with basic algebras

Margolis

Steinberg

2012

Compositio Math.

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“…We follow the ideas of [2], where the same procedure was applied to the Hopf algebra of Malvenuto and Reutenauer. The same Möbius inversion trick has been employed frequently in inverse semigroup representation theory with a variety of purposes [19,[26][27][28][29]. For each element g ∈ P n let…”

Section: Remark 42mentioning

confidence: 99%

“…This paper explores one aspect of the relationship between Schur-Weyl duality, diagram algebras, and inverse semigroups whose general study was started by Solomon [27] and continued by several authors [13][14][15]19]. We thank the referees for bringing up these and other references to our attention.…”

Section: Introductionmentioning

confidence: 97%

The Hopf algebra of uniform block permutations

Aguiar

Orellana

2008

J Algebr Comb

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We introduce the Hopf algebra of uniform block permutations and show that it is self-dual, free, and cofree. These results are closely related to the fact that uniform block permutations form a factorizable inverse monoid. This Hopf algebra contains the Hopf algebra of permutations of Malvenuto and Reutenauer and the Hopf algebra of symmetric functions in non-commuting variables of Gebhard, Rosas, and Sagan. These two embeddings correspond to the factorization of a uniform block permutation as a product of an invertible element and an idempotent one.

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“…The reason for the name is due to a connection with the so-called rook monoids (and associated algebras and deformations) studied by Halverson, Solomon and others [12,18,32,34,36,68,72]. As noted by Grood [33], Solomon's discovery [72] of a Schur-Weyl duality for rook monoid algebras (see also [50]) led to the investigation of a number of other "rook diagram algebras", such the rook Brauer algebras [35,60], Motzkin algebras [5] and more. Such studies, and other considerations often to do with representation theory and/or statistical mechanics, have led to the discovery and investigation of a great many other families of diagram algebras [6,10,11,45,61,62,67,69].…”

Section: Introductionmentioning

confidence: 99%