Representations of rational Cherednik algebras in positive characteristic

Balagović, Martina; Chen, Harrison

doi:10.1016/j.jpaa.2012.09.015

Cited by 16 publications

(32 citation statements)

References 20 publications

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“…Indeed, in the modular setting, the Hochschild cohomology of the acting group algebra is nontrivial and thus gives rise to deformations of the skew group algebra of a completely new flavor. Study has recently begun on the representation theory of deformations of skew group algebras in positive characteristic (see [1,2,3,6,21] for example) and these new types of deformations may help in developing a unified theory. In addition to combinatorial tools, we rely on homological algebra to reveal and to understand these new types of deformations that do not appear in characteristic zero.…”

Section: Introductionmentioning

confidence: 99%

PBW Deformations of Skew Group Algebras in Positive Characteristic

Shepler

Witherspoon

2014

Algebr Represent Theor

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Abstract. We investigate deformations of a skew group algebra that arise from a finite group acting on a polynomial ring. When the characteristic of the underlying field divides the order of the group, a new type of deformation emerges that does not occur in characteristic zero. This analogue of Lusztig's graded affine Hecke algebra for positive characteristic can not be forged from the template of symplectic reflection and related algebras as originally crafted by Drinfeld. By contrast, we show that in characteristic zero, for arbitrary finite groups, a Lusztig-type deformation is always isomorphic to a Drinfeldtype deformation. We fit all these deformations into a general theory, connecting Poincaré-Birkhoff-Witt deformations and Hochschild cohomology when working over fields of arbitrary characteristic. We make this connection by way of a double complex adapted from Guccione, Guccione, and Valqui, formed from the Koszul resolution of a polynomial ring and the bar resolution of a group algebra.

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Section: Introductionmentioning

confidence: 99%

PBW Deformations of Skew Group Algebras in Positive Characteristic

Shepler

Witherspoon

2014

Algebr Represent Theor

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“…In this article we continue the study, initiated in [6], [7], [2] and [1], of these algebras at t = 1 and over a field of positive characteristic. We focus on the representation theoretic aspects of the story.…”

Section: Rational Cherednik Algebras Were Introduced By Etingof and Gmentioning

confidence: 81%

“…Therefore, we assume throughout this section that p, the characteristic of k, is coprime to |W |. There is a different presentation, given in [2], of the rational Cherednik algebra which is valid when W contains transvection.…”

Section: Rational Cherednik Algebrasmentioning

confidence: 99%

On the Smoothness of Centres of Rational Cherednik Algebras in Positive Characteristic

Bellamy

Martino

2013

Glasgow Math. J.

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Abstract. In this article we study rational Cherednik algebras at t = 1 in positive characteristic. We study a finite dimensional quotient of the rational Cherednik algebra called the restricted rational Cherednik algebra. When the corresponding pseudo-reflection group belongs to the infinite series G(m, d, n), we describe explicitly the block decomposition of the restricted algebra. We also classify all pseudo-reflection groups for which the centre of the corresponding rational Cherednik algebra is regular for generic values of the deformation parameter.

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“…This explains why we examined the Hilbert series of Q G in our context. In fact, rational Cherednik algebras H c (G) for G = GL n (F q ) and their finite dimensional representations L c (triv) have been studied by Balagović and Chen [4]. However, their results show that the common kernel of the Dunkl operators in H c (G) acting on S = F q [x] is not spanned by x q n 1 , .…”

Section: Further Questions and Remarksmentioning

confidence: 99%