scholarly journals Parameters for generalized Hecke algebras in type B

2019 ◽  
Vol 18 (09) ◽  
pp. 1950173
Author(s):  
Max Murin ◽  
Seth Shelley-Abrahamson
Keyword(s):  
Recent Work ◽  
Parabolic Subgroup ◽  
Coxeter Group ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Irreducible Representations ◽  
Type B ◽  
Full Support

The irreducible representations of full support in the rational Cherednik category [Formula: see text] attached to a Coxeter group [Formula: see text] are in bijection with the irreducible representations of an associated Iwahori–Hecke algebra. Recent work has shown that the irreducible representations in [Formula: see text] of arbitrary given support are similarly governed by certain generalized Hecke algebras. In this paper, we compute the parameters for these generalized Hecke algebras in the remaining previously unknown cases, corresponding to the parabolic subgroup [Formula: see text] in [Formula: see text] for [Formula: see text] and [Formula: see text].

0-Hecke algebras

1979 ◽  
Vol 27 (3) ◽  
pp. 337-357 ◽  
Author(s):  
P. N. Norton
Keyword(s):  
Coxeter Group ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Irreducible Representations ◽  
One Dimensional ◽  
Direct Sum ◽  
Natural Way

AbstractThe structure of a 0-Hecke algebra H of type (W, R) over a field is examined. H has 2n distinct irreducible representations, where n = ∣R∣, all of which are one-dimensional, and correspond in a natural way with subsets of R. H can be written as a direct sum of 2n indecomposable left ideals, in a similar way to Solomon's (1968) decomposition of the underlying Coxeter group W.

The boundedness of a weighted Coxeter group with non-3-edge-labeling graph

2019 ◽  
Vol 18 (05) ◽  
pp. 1950085
Author(s):  
Yan Li ◽  
Jian-Yi Shi
Keyword(s):  
Parabolic Subgroup ◽  
Coxeter Group ◽  
Upper Bound ◽  
Hecke Algebras ◽  
Product Formula ◽  
Edge Labeling

Let [Formula: see text] be a weighted Coxeter group such that the order [Formula: see text] of the product [Formula: see text] is not 3 for any [Formula: see text] and that [Formula: see text], where [Formula: see text] is the longest element in the parabolic subgroup [Formula: see text] of [Formula: see text] generated by [Formula: see text]. We prove that [Formula: see text] is bounded with [Formula: see text] an upper bound in the sense of Lusztig in Sec. 13.2 of [Hecke Algebras with Unequal Parameters, arXiv:math/0208154 v2 [math.RT] 10 Jun 2014], verifying a conjecture of Lusztig in our case (see Conjecture 13.4 in loc. cite).

scholarly journals A decomposition formula for representations

1987 ◽  
Vol 107 ◽  
pp. 63-68 ◽  
Author(s):  
George Kempf
Keyword(s):  
Irreducible Representation ◽  
General Formula ◽  
Parabolic Subgroup ◽  
Characteristic Zero ◽  
Maximal Torus ◽  
Reductive Group ◽  
Irreducible Representations ◽  
Levi Subgroup ◽  
Know How ◽  
Decomposition Formula

Let H be the Levi subgroup of a parabolic subgroup of a split reductive group G. In characteristic zero, an irreducible representation V of G decomposes when restricted to H into a sum V = ⊕mαWα where the Wα’s are distinct irreducible representations of H. We will give a formula for the multiplicities mα. When H is the maximal torus, this formula is Weyl’s character formula. In theory one may deduce the general formula from Weyl’s result but I do not know how to do this.

scholarly journals A quantum probabilistic approach to Hecke algebras for 𝔭-adic PGL2

2018 ◽  
Vol 21 (03) ◽  
pp. 1850015
Author(s):  
Takehiro Hasegawa ◽  
Hayato Saigo ◽  
Seiken Saito ◽  
Shingo Sugiyama
Keyword(s):  
Inversion Formula ◽  
Probabilistic Approach ◽  
Hecke Algebras ◽  
Hecke Algebra ◽  
Quantum Probability ◽  
Probabilistic Interpretation ◽  
Fourier Inversion ◽  
Fourier Inversion Formula ◽  
The Subject

The subject of the present paper is an application of quantum probability to [Formula: see text]-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for [Formula: see text], where [Formula: see text] is a [Formula: see text]-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for [Formula: see text].

scholarly journals The representation type of Hecke algebras of type B

2004 ◽  
Vol 181 (1) ◽  
pp. 134-159 ◽  
Author(s):  
Susumu Ariki ◽  
Andrew Mathas
Keyword(s):  
Hecke Algebras ◽  
Representation Type ◽  
Type B

scholarly journals Mixed Perverse Sheaves on Flag Varieties for Coxeter Groups

2019 ◽  
Vol 72 (1) ◽  
pp. 1-55
Author(s):  
Pramod N. Achar ◽  
Simon Riche ◽  
Cristian Vay
Keyword(s):  
Coxeter Group ◽  
Coxeter Groups ◽  
Grothendieck Group ◽  
General Setting ◽  
Hecke Algebra ◽  
Abelian Category ◽  
Flag Varieties ◽  
Perverse Sheaves

AbstractIn this paper we construct an abelian category of mixed perverse sheaves attached to any realization of a Coxeter group, in terms of the associated Elias–Williamson diagrammatic category. This construction extends previous work of the first two authors, where we worked with parity complexes instead of diagrams, and we extend most of the properties known in this case to the general setting. As an application we prove that the split Grothendieck group of the Elias–Williamson diagrammatic category is isomorphic to the corresponding Hecke algebra, for any choice of realization.

scholarly journals Leading coefficients and cellular bases of Hecke algebras

2009 ◽  
Vol 52 (3) ◽  
pp. 653-677 ◽  
Author(s):  
Meinolf Geck
Keyword(s):  
Weyl Group ◽  
Coxeter Group ◽  
Cellular Structure ◽  
Hecke Algebras ◽  
Point Of View ◽  
Reflection Groups ◽  
Cellular Basis ◽  
Complex Reflection Groups ◽  
Finite Coxeter Group ◽  
New Construction

AbstractLet H be the generic Iwahori–Hecke algebra associated with a finite Coxeter group W. Recently, we have shown that H admits a natural cellular basis in the sense of Graham and Lehrer, provided that W is a Weyl group and all parameters of H are equal. The construction involves some data arising from the Kazhdan–Lusztig basis {Cw} of H and Lusztig's asymptotic ring J}. We attempt to study J and its representation theory from a new point of view. We show that J can be obtained in an entirely different fashion from the generic representations of H, without any reference to {Cw}. We then extend the construction of the cellular basis to the case where W is not crystallographic. Furthermore, if H is a multi-parameter algebra, we see that there always exists at least one cellular structure on H. Finally, the new construction of J may be extended to Hecke algebras associated with complex reflection groups.

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