Affine Weyl groups and affine Hecke algebras

1994 ◽  
pp. 18-26
Author(s):  
Nanhua Xi
Keyword(s):  
Hecke Algebras ◽  
Weyl Groups ◽  
Affine Weyl Groups ◽  
Affine Hecke Algebras

scholarly journals Metaplectic representations of Hecke algebras, Weyl group actions, and associated polynomials

2021 ◽  
Vol 27 (3) ◽  
Author(s):  
Siddhartha Sahi ◽  
Jasper V. Stokman ◽  
Vidya Venkateswaran
Keyword(s):  
Weyl Group ◽  
Hecke Algebras ◽  
Principal Series ◽  
Rational Functions ◽  
Algebraic Combinatorics ◽  
Weyl Groups ◽  
Whittaker Functions ◽  
Double Affine Hecke Algebra ◽  
Associated Polynomials ◽  
Affine Hecke Algebras

AbstractWe construct a family of representations of affine Hecke algebras, which depend on a number of auxiliary parameters $$g_i$$ g i , and which we refer to as metaplectic representations. We realize these representations as quotients of certain parabolically induced modules, and we apply the method of Baxterization (localization) to obtain actions of corresponding Weyl groups on rational functions on the torus. Our construction both generalizes and provides a conceptual proof of earlier results of Chinta, Gunnells, and Puskas, which had depended on a crucial computer verification. A key motivation is that when the parameters $$g_i$$ g i are specialized to certain Gauss sums, the resulting representation and its localization arise naturally in the consideration of p-parts of Weyl group multiple Dirichlet series. In this special case, similar results have been previously obtained in the literature by the study of Iwahori Whittaker functions for principal series of metaplectic covers of reductive p-adic groups. However this technique is not available for generic parameters $$g_i$$ g i . It turns out that the metaplectic representations can be extended to the double affine Hecke algebra, where they share many important properties with Cherednik’s basic polynomial representation, which they generalize. This allows us to introduce families of metaplectic polynomials, which depend on the $$g_i$$ g i , and which generalize Macdonald polynomials. In this paper we discuss in some detail the situation for type A, which is of considerable interest in algebraic combinatorics. We postpone some of the proofs, as well as a discussion of other types, to the sequel.

scholarly journals Solution of tetrahedron equation and cluster algebras

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
P. Gavrylenko ◽  
M. Semenyakin ◽  
Y. Zenkevich
Keyword(s):  
Integrable System ◽  
Cluster Algebra ◽  
Newton Polygon ◽  
Cluster Algebras ◽  
Weyl Groups ◽  
Newton Polygons ◽  
Tetrahedron Equation ◽  
Affine Weyl Groups ◽  
New Formalism ◽  
Bruhat Cell

Abstract We notice a remarkable connection between the Bazhanov-Sergeev solution of Zamolodchikov tetrahedron equation and certain well-known cluster algebra expression. The tetrahedron transformation is then identified with a sequence of four mutations. As an application of the new formalism, we show how to construct an integrable system with the spectral curve with arbitrary symmetric Newton polygon. Finally, we embed this integrable system into the double Bruhat cell of a Poisson-Lie group, show how triangular decomposition can be used to extend our approach to the general non-symmetric Newton polygons, and prove the Lemma which classifies conjugacy classes in double affine Weyl groups of A-type by decorated Newton polygons.

scholarly journals Rouquier blocks of the cyclotomic Hecke algebras of G(de, e, r)

2010 ◽  
Vol 197 ◽  
pp. 175-212
Author(s):  
Maria Chlouveraki
Keyword(s):  
Infinite Series ◽  
Hecke Algebras ◽  
Weyl Groups ◽  
Reflection Groups ◽  
Complex Reflection ◽  
Complex Reflection Groups ◽  
Cyclotomic Hecke Algebras

The Rouquier blocks of the cyclotomic Hecke algebras, introduced by Rouquier, are a substitute for the families of characters defined by Lusztig for Weyl groups, which can be applied to all complex reflection groups. In this article, we determine them for the cyclotomic Hecke algebras of the groups of the infinite seriesG(de, e, r), thus completing their calculation for all complex reflection groups.

scholarly journals Bijective projections on parabolic quotients of affine Weyl groups

2014 ◽  
Vol 41 (4) ◽  
pp. 911-948 ◽  
Author(s):  
Elizabeth Beazley ◽  
Margaret Nichols ◽  
Min Hae Park ◽  
XiaoLin Shi ◽  
Alexander Youcis
Keyword(s):  
Weyl Groups ◽  
Affine Weyl Groups
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