Induced Representations of Affine Hecke Algebras and Canonical Bases of Quantum Groups

Leclerc, Bernard; Nazarov, Maxim; Thibon, Jean‐Yves

doi:10.1007/978-1-4612-0045-1_6

Cited by 44 publications

(67 citation statements)

References 28 publications

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“…We prove that they also belong to S * . Hence using [33,49], it follows that S * ∩ B * contains all multiplicative products of flag minors. However the two bases differ in general.…”

Section: 4mentioning

confidence: 99%

Semicanonical bases and preprojective algebras

Geiß¹,

Leclerc²,

Schröer³

2005

Annales Scientifiques de l’École Normale Supérieure

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ABSTRACT. We study the multiplicative properties of the dual of Lusztig's semicanonical basis. The elements of this basis are naturally indexed by the irreducible components of Lusztig's nilpotent varieties, which can be interpreted as varieties of modules over preprojective algebras. We prove that the product of two dual semicanonical basis vectors ρ Z ′ and ρ Z ′′ is again a dual semicanonical basis vector provided the closure of the direct sum of the corresponding two irreducible components Z ′ and Z ′′ is again an irreducible component. It follows that the semicanonical basis and the canonical basis coincide if and only if we are in Dynkin type An with n ≤ 4. Finally, we provide a detailed study of the varieties of modules over the preprojective algebra of type A5. We show that in this case the multiplicative properties of the dual semicanonical basis are controlled by the Ringel form of a certain tubular algebra of type (6, 3, 2) and by the corresponding elliptic root system of type E (1,1) 8 . RÉSUMÉ. Nousétudions les propriétés multiplicatives de la base duale de la base semi-canonique de Lusztig. Leséléments de cette base sont naturellement paramétrés par les composantes irréduc-tibles des variétés nilpotentes de Lusztig, qui peuventêtre interprétées comme variétés de modules sur les algèbres préprojectives. Nous démontrons que le produit de deux vecteurs ρ Z ′ et ρ Z ′′ de la base semi-canonique duale est encore un vecteur de la base semi-canonique duale si la somme directe des composantes irréductibles Z ′ et Z ′′ est encore une composante irréductible. Il en résulte que les bases canonique et semi-canonique ne coïncident que pour le type de Dynkin An avec n ≤ 4. Finalement, nousétudions en détail les variétés de modules sur l'algèbre préprojective de type A5. Nous montrons que dans ce cas les propriétés multiplicatives de la base semi-canonique duale sont controllées par la forme de Ringel d'une algèbre tubulaire de type (6, 3, 2) et par le système de racines elliptique de type E (1,1) 8 qui lui est associé.

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“…We prove that they also belong to S * . Hence using [33,49], it follows that S * ∩ B * contains all multiplicative products of flag minors. However the two bases differ in general.…”

Section: 4mentioning

confidence: 99%

Semicanonical bases and preprojective algebras

Geiß¹,

Leclerc²,

Schröer³

2005

Annales Scientifiques de l’École Normale Supérieure

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“…Since then these connections have been studied by several authors and other similar correspondences have been discovered [A,LT2,VV,Gr,BK,LNT,B1,B2].…”

Section: Introductionmentioning

confidence: 99%

Dual canonical bases, quantum shuffles and q -characters

Leclerc

2004

Mathematische Zeitschrift

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132

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Rosso and Green have shown how to embed the positive part U q (n) of a quantum enveloping algebra U q (g) in a quantum shuffle algebra. In this paper we study some properties of the image of the dual canonical basis B * of U q (n) under this embedding . This is motivated by the fact that when g is of type A r , the elements of (B * ) are q-analogues of irreducible characters of the affine Iwahori-Hecke algebras attached to the groups GL(m) over a p-adic field.

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“…Recall that i(Q) = (2, 1, 4, 3, 2, 1, 5, 4, 3, 2, 1, 5, 4, 3, 5). Table 1 Inequality of L st (Q) Inequality of L PBW (Q) a 2 + a 4 a 1 + a 5 c 11 c 22 a 5 a 2 + a 6 c 22 + c 23 c 11 + c 12 a 4 + a 7 a 3 + a 8 c 23 + c 13 c 34 + c 24 a 5 + a 8 a 4 + a 9 c 22 c 33 a 6 + a 9 a 5 + a 10 c 33 + c 34 c 22 + c 23 a 10 a 6 + a 11 c 24 + c 25 c 13 + c 14 a 8 a 7 + a 12 c 34 + c 24 c 45 + c 35 a 9 + a 12 a 8 + a 13 c 33 c 44 a 10 + a 13 a 9 + a 14 c 44 + c 45 c 33 + c 34 a 13 a 12 + a 15 c 44 c 55…”

Section: Inequality (C) Comes From (L1) and Inequalities (D) Come Frmentioning

confidence: 97%

Canonical basis linearity regions arising from quiver representations

Marsh

Reineke

2003

Journal of Algebra

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In this paper we show that there is a link between the combinatorics of the canonical basis of a quantized enveloping algebra and the monomial bases of the second author [Math. Z. 237 (2001) 639] arising from representations of quivers. We prove that some reparametrization functions of the canonical basis, arising from the link between Lusztig's approach to the canonical basis and the string parametrization of the canonical basis, are given on a large cone by linear functions arising from these monomial bases for a quantized enveloping algebra.

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Induced Representations of Affine Hecke Algebras and Canonical Bases of Quantum Groups

Abstract: A criterion of irreducibility for induction products of evaluation modules of type A affine Hecke algebras is given. It is derived from multiplicative properties of the canonical basis of a quantum deformation of the Bernstein-Zelevinsky ring.

Cited by 44 publications

References 28 publications

Semicanonical bases and preprojective algebras

Semicanonical bases and preprojective algebras

Dual canonical bases, quantum shuffles and q -characters

Canonical basis linearity regions arising from quiver representations

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