Tableaux Realization of Generalized Verma Modules

Mazorchuk, Volodymyr

doi:10.4153/cjm-1998-043-x

Cited by 23 publications

(14 citation statements)

References 8 publications

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“…This theory was generalized to orthogonal Lie algebras in [Ma3] and to quantum algebras in [MT]. It was also very useful for the study of various categories of U n -modules, see [Kh,Ma1,Ma4 A major step in the development of the theory of Gelfand-Zeitlin modules was made in [Ov1,Ov2] (it was also put into a more general setup in [FO], see also [FMO, FOS]) where it was, in particular, shown that all Gelfand-Zeitlin characters 1 The surname Zeitlin appears in the literature in different spellings, that is in different latinizations of the Cyrillic version of the original Latin (German) surname, in particular, it was spelled as Cetlin, Zetlin, Tzetlin and Tsetlin. Here we use the original Latin spelling.…”

Section: Introduction and Description Of The Resultsmentioning

confidence: 99%

Canonical Gelfand–Zeitlin Modules over Orthogonal Gelfand–Zeitlin Algebras

Early

Mazorchuk

Vishnyakova

2019

International Mathematics Research Notices

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We prove that every orthogonal Gelfand-Zeitlin algebra U acts (faithfully) on its Gelfand-Zeitlin subalgebra Γ. Considering the dual module, we show that every Gelfand-Zeitlin character of Γ is realizable in a U -module. We observe that the Gelfand-Zeitlin formulae can be rewritten using divided difference operators. It turns out that the action of the latter operators on Γ gives rise to an explicit basis in a certain Gelfand-Zeitlin submodule of the dual module mentioned above. This gives, generically, both in the case of regular and singular Gelfand-Zeitlin characters, an explicit construction of simple modules which realize the given Gelfand-Zeitlin characters.

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Section: Introduction and Description Of The Resultsmentioning

confidence: 99%

Canonical Gelfand–Zeitlin Modules over Orthogonal Gelfand–Zeitlin Algebras

Early

Mazorchuk

Vishnyakova

2019

International Mathematics Research Notices

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“…One of the main advantages of Gelfand-Zetlin modules is that many Gelfand-Zetlin modules admit a so-called tableau realization, that is an explicit combinatorial construction in which a basis of the module is indexed by a set of tableaux (defined by imposing some conditions on the entries) and the action of the generators e i,i+1 and e i+1,i of g is given by the so-called Gelfand-Zetlin formulae, see [DFO], [Maz1], [Maz2] for details. In such a realization the tableaux represents a basis of common eigenvectors for Γ (as mentioned above, the corresponding eigenvalues are computed as certain symmetric functions), and the action of e i,i+1 (resp.…”

Section: Gelfand-zetlin Realizationmentioning

confidence: 99%

Cuspidalsln-modules and deformations of certain Brauer tree algebras

Mazorchuk

Stroppel

2011

Advances in Mathematics

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We show that the algebras describing blocks of the category of cuspidal weight (respectively generalized weight) sln-modules are one-parameter (respectively multi-parameter) deformations of certain Brauer tree algebras. We explicitly determine these deformations both graded and ungraded. The algebras we deform also appear as special centralizer subalgebras of Temperley-Lieb algebras or as generalized Khovanov algebras. They show up in the context of highest weight representations of the Virasoro algebra, in the context of rational representations of the general linear group and Schur algebras and in the study of the Milnor fiber of Kleinian singularities.

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“…It is related to many concepts arizing in Mathematics and Physics, see for example [1,2,4,9,[15][16][17]19,20]. The general theory of Gelfand-Tsetlin modules for gl n was developed in [3,[5][6][7]10,22,23,25,26], and references therein. For U q (gl n ) certain families of Gelfand-Tsetlin modules were constructed in [12,24], while the general theory was developed in [8].…”

Section: Introductionmentioning

confidence: 99%