Affine weyl groups and conjugacy classes in Weyl groups

1996 ◽  
Vol 1 (1-2) ◽  
pp. 83-97 ◽  
Author(s):  
George Lusztig
Keyword(s):  
Conjugacy Classes ◽  
Weyl Groups ◽  
Affine Weyl Groups

scholarly journals Minimal length elements of extended affine Weyl groups

2014 ◽  
Vol 150 (11) ◽  
pp. 1903-1927 ◽  
Author(s):  
Xuhua He ◽  
Sian Nie
Keyword(s):  
Conjugacy Class ◽  
Crucial Role ◽  
Minimal Length ◽  
Conjugacy Classes ◽  
Weyl Groups ◽  
Affine Weyl Groups ◽  
Affine Hecke Algebra ◽  
Affine Weyl Group ◽  
Twisted Conjugacy ◽  
Class Polynomials

AbstractLet $W$ be an extended affine Weyl group. We prove that the minimal length elements $w_{{\mathcal{O}}}$ of any conjugacy class ${\mathcal{O}}$ of $W$ satisfy some nice properties, generalizing results of Geck and Pfeiffer [On the irreducible characters of Hecke algebras, Adv. Math. 102 (1993), 79–94] on finite Weyl groups. We also study a special class of conjugacy classes, the straight conjugacy classes. These conjugacy classes are in a natural bijection with the Frobenius-twisted conjugacy classes of some $p$-adic group and satisfy additional interesting properties. Furthermore, we discuss some applications to the affine Hecke algebra $H$. We prove that $T_{w_{{\mathcal{O}}}}$, where ${\mathcal{O}}$ ranges over all the conjugacy classes of $W$, forms a basis of the cocenter $H/[H,H]$. We also introduce the class polynomials, which play a crucial role in the study of affine Deligne–Lusztig varieties He [Geometric and cohomological properties of affine Deligne–Lusztig varieties, Ann. of Math. (2) 179 (2014), 367–404].

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 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

scholarly journals Tight Quotients and Double Quotients in the Bruhat Order

10.37236/1871 ◽  
2005 ◽  
Vol 11 (2) ◽  
Author(s):  
John R. Stembridge
Keyword(s):  
Coxeter Groups ◽  
Upper Bounds ◽  
Bruhat Order ◽  
Weyl Groups ◽  
Order Dimension ◽  
Parabolic Subgroups ◽  
Affine Weyl Groups ◽  
Positive Side ◽  
Maximal Parabolic Subgroups

It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly, the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided (i.e., "double") quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation, and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases, these orbit representations are "tight" in the sense that the root system induces an ordering on the orbit that yields effective coordinates for the Bruhat order, and hence also provides upper bounds for the order dimension. In this paper, we (1) provide a general characterization of tightness for one-sided quotients, (2) classify all tight one-sided quotients of finite Coxeter groups, and (3) classify all tight double quotients of affine Weyl groups.

scholarly journals Extended Affine Weyl Groups: Presentation by Conjugation via Integral Collection

2011 ◽  
Vol 39 (2) ◽  
pp. 730-749 ◽  
Author(s):  
Saeid Azam ◽  
Valiollah Shahsanaei
Keyword(s):  
Weyl Groups ◽  
Affine Weyl Groups

scholarly journals Poincaré duality and Langlands duality for extended affine Weyl groups

2018 ◽  
Vol 3 (3) ◽  
pp. 491-522
Author(s):  
Graham Niblo ◽  
Roger Plymen ◽  
Nick Wright
Keyword(s):  
Weyl Groups ◽  
Poincaré Duality ◽  
Poincare Duality ◽  
Affine Weyl Groups ◽  
Langlands Duality

scholarly journals Extended affine Weyl groups of BCD-type: Their Frobenius manifolds and Landau–Ginzburg superpotentials

2019 ◽  
Vol 351 ◽  
pp. 897-946 ◽  
Author(s):  
Boris Dubrovin ◽  
Ian A.B. Strachan ◽  
Youjin Zhang ◽  
Dafeng Zuo
Keyword(s):  
Weyl Groups ◽  
Affine Weyl Groups ◽  
Frobenius Manifolds
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