Singular Gelfand–Tsetlin modules of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:mi mathvariant="fraktur">gl</mml:mi></mml:mrow><mml:mo stretchy="false">(</mml:mo><mml:mi>n</mml:mi><mml:mo stretchy="false">)</mml:mo></mml:math>

Futorny, Vyacheslav; Grantcharov, Dimitar; Ramirez, Luis Enrique

doi:10.1016/j.aim.2015.12.001

Cited by 40 publications

(73 citation statements)

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“…In particular, our method works for q = 1 providing new families of irreducible Gelfand-Tsetlin modules for gl n . This generalizes the results of [10] and [15]. …”

supporting

confidence: 65%

“…In [10] the classical Gelfand-Tsetlin formulas were generalized allowing to construct a new family of gl n -modules associated with tableaux with at most one singular pair. In this section we will use this approach and combine with the ideas of Section 3 to construct new families of U q -modules.…”

Section: New Gelfand-tsetlin Modulesmentioning

confidence: 99%

“…Again, these modules allow arbitrary singularities (generalizing [10]) and on the other hand have Gelfand-Tsetlin multiplicities bounded by 2 with a non-diagonalizable action of the Gelfand-Tsetlin subalgebra (generalizing [15]). Moreover, any irreducible Gelfand-Tsetlin module with a designated singularity appear as a subquotient of the constructed universal module (Theorem 5.2).…”

Section: Introductionmentioning

confidence: 99%

“…Further, infinite dimensional GelfandTsetlin modules for gl n were studied in [18], [25], [26], [3], [31], [7], [27], [28], [30], [8], [9], [10], [11], [12], [37], [32], [35], [36] among the others. These representations have close connections to different concepts in Mathematics and Physics (cf.…”

Section: Introductionmentioning

confidence: 99%

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Gelfand–Tsetlin modules of quantum gln defined by admissible sets of relations

2018

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Abstract. The purpose of this paper is to construct new families of irreducible Gelfand-Tsetlin modules for Uq(gl n ). These modules have arbitrary singularity and Gelfand-Tsetlin multiplicities bounded by 2. Most previously known irreducible modules had all Gelfand-Tsetlin multiplicities bounded by 1 [16], [17]. In particular, our method works for q = 1 providing new families of irreducible Gelfand-Tsetlin modules for gl n . This generalizes the results of [10] and [15].

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“…In particular, our method works for q = 1 providing new families of irreducible Gelfand-Tsetlin modules for gl n . This generalizes the results of [10] and [15]. …”

supporting

confidence: 65%

Section: New Gelfand-tsetlin Modulesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Gelfand–Tsetlin modules of quantum gln defined by admissible sets of relations

2018

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“…An explicit construction of such modules was given in [8]. In particular, we show that the modules constructed in [8] exhaust all irreducible Gelfand-Tsetlin modules with 1-singularity. To prove the result we introduce a new category of modules (called Drinfeld category) related to the Drinfeld generators of the Yangian Y (gl n ) and define a functor from the category of non-critical Gelfand-Tsetlin modules to the Drinfeld category.1991 Mathematics Subject Classification.…”

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confidence: 89%

Drinfeld category and the classification of singular Gelfand–Tsetlin $\displaystyle \mathfrak{gl}_n$-modules

Futorny

Grantcharov

Ramirez

2017

International Mathematics Research Notices

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We prove a uniqueness theorem for irreducible non-critical Gelfand-Tsetlin modules. The uniqueness result leads to a complete classification of the irreducible Gelfand-Tsetlin modules with 1-singularity. An explicit construction of such modules was given in [8]. In particular, we show that the modules constructed in [8] exhaust all irreducible Gelfand-Tsetlin modules with 1-singularity. To prove the result we introduce a new category of modules (called Drinfeld category) related to the Drinfeld generators of the Yangian Y (gl n ) and define a functor from the category of non-critical Gelfand-Tsetlin modules to the Drinfeld category.1991 Mathematics Subject Classification. Primary 17B67.Henceforth, we define V (L) to be the complex vector space with basis the set L Z , i.e.An open problem which dates back to the work of Gelfand and Graev, [11], is to define a gl n -module structure on a subspace of V (L) annihilated by some maximal ideal of the Gelfand-Tsetlin subalgebra. Solutions to this problem are known in various cases, the most fundamental of which relies on the construction of Gelfand-Tsetlin bases of finite dimensional representations of g.Another important solution concerns the generic Gelfand-Tsetlin modules. More precisely, a pair of entries (l mi , l mj ) of L such that l mi − l mj ∈ Z is called a singular pair. If T (L) contains a singular pair in the m-th row for some 2 ≤ m ≤ n − 1 then we call T (L) a singular tableau (and L a singular element in T n (C)). A tableau T (L) is 1-singular if it contains exactly one singular pair. If T (L) is not singular it is called generic. For a generic tableau T (L), one can imitate the construction of Gelfand-Tsetlin bases of finite-dimensional representations and construct a gl nmodule structure on V (L), see for example [5].The study of singular modules V (L) was initiated in [8], where a module structure on V (L) was introduced for any 1-singular tableau L. The latter construction was generalized in [9] for singular tableaux L with multiple singular pairs such that the difference of the entries of any two distinct singular pairs is noniteger. By taking irreducible quotients of V (L) new irreducible Gelfand-Tsetlin modules for gl n were constructed. In particular, understanding the 1-singular case helped us to complete classification of irreducible Gelfand-Tsetlin gl 3 -modules in [10]. Explicit basis for a class of irreducible 1-singular Gelfand-Tsetlin gl n -modules was constructed in [13].We note that to each L ∈ T n (C) we associate the maximal ideal m L of the Gelfand-Tsetlin subalgebra Γ generated by c ij −γ ij (L), 1 ≤ j ≤ i ≤ n, where c ij are the generators of Γ, and γ ij are symmetric polynomials defined in (3). Furthermore, for every maximal ideal m of Γ there exists an irreducible Gelfand-Tsetlin module V such that V m = 0 (see (2) for the definition of V m ) and the number of such non-isomorphic irreducible modules V is finite, [24]. In fact, for a generic L there exists a unique up to isomorphism irreducible module V such that V mL = 0. On the othe...

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Weight Representations of Admissible Affine Vertex Algebras

Arakawa

Futorny

Ramirez

2017

Commun. Math. Phys.

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For an admissible affine vertex algebra V k (g) of type A, we describe a new family of relaxed highest weight representations of V k (g). They are simple quotients of representations of the affine Kac-Moody algebra g induced from the following g-modules: 1) generic Gelfand-Tsetlin modules in the principal nilpotent orbit, in particular all such modules induced from sl 2 ; 2) all Gelfand-Tsetlin modules in the principal nilpotent orbit which are induced from sl 3 ; 3) all simple Gelfand-Tsetlin modules over sl 3 . This in particular gives the classification of all simple positive energy weight representations of V k (g) with finite dimensional weight spaces for g = sl 3 .

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Singular Gelfand–Tsetlin modules of gl(n)

Cited by 40 publications

References 23 publications

Gelfand–Tsetlin modules of quantum gln defined by admissible sets of relations

Gelfand–Tsetlin modules of quantum gln defined by admissible sets of relations

Drinfeld category and the classification of singular Gelfand–Tsetlin $\displaystyle \mathfrak{gl}_n$-modules

Weight Representations of Admissible Affine Vertex Algebras

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