INVERSIONS IN CLASSICAL WEYL GROUPS

2007 ◽  
Vol 09 (01) ◽  
pp. 1-20
Author(s):  
KEQUAN DING ◽  
SIYE WU
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Root Systems ◽  
Flag Manifold ◽  
Length Function ◽  
Weyl Groups ◽  
Flag Manifolds ◽  
A Finite Group

We introduce inversions for classical Weyl group elements and relate them, by counting, to the length function, root systems and Schubert cells in flag manifolds. Special inversions are those that only change signs in the Weyl groups of types Bn, Cnand Dn. Their counting is related to the (only) generator of the Weyl group that changes signs, to the corresponding roots, and to a special subvariety in the flag manifold fixed by a finite group.

WEYL GROUPS AND BASIS ELEMENTS OF HECKE ALGEBRAS OF GELFAND–GRAEV REPRESENTATIONS

2011 ◽  
Vol 10 (05) ◽  
pp. 849-864 ◽  
Author(s):  
JULIANNE G. RAINBOLT
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Hecke Algebras ◽  
Initial Section ◽  
Weyl Groups ◽  
Group Of Lie Type ◽  
Combinatorial Description ◽  
Algorithmic Method ◽  
A Finite Group ◽  
Theoretic Description

The initial section of this article provides illustrative examples on two ways to construct the Weyl group of a finite group of Lie type. These examples provide the background for a comparison of the elements in the Weyl groups of GL(n, q) and U(n, q) that are used in the construction of the standard bases of the Hecke algebras of the Gelfand–Graev representations of GL(n, q) and U(n, q). Using a theorem of Steinberg, a connection between a theoretic description of bases of these Hecke algebras and a combinatorial description of these bases is provided. This leads to an algorithmic method for generating bases of the Hecke algebras of the Gelfand–Graev representations of GL(n, q) and U(n, q).

scholarly journals Cohomology of complements of toric arrangements associated with root systems

2022 ◽  
Vol 9 (1) ◽  
Author(s):  
Olof Bergvall
Keyword(s):  
Finite Group ◽  
Root Systems ◽  
Important Application ◽  
Weyl Groups ◽  
Cohomology Groups ◽  
A Finite Group

AbstractWe develop an algorithm for computing the cohomology of complements of toric arrangements. In the case a finite group $$\Gamma $$ Γ is acting on the arrangement, the algorithm determines the cohomology groups as representations of $$\Gamma $$ Γ . As an important application, we determine the cohomology groups of the complements of the toric arrangements associated with root systems of exceptional type as representations of the corresponding Weyl groups.

scholarly journals A Length Function for Weyl Groups of Extended Affine Root Systems of Type A1

2015 ◽  
Vol 22 (04) ◽  
pp. 621-638 ◽  
Author(s):  
Saeid Azam ◽  
Mohammad Nikouei
Keyword(s):  
Weyl Group ◽  
Root Systems ◽  
Type A ◽  
Length Function ◽  
Weyl Groups ◽  
Combinatorial Properties ◽  
Affine Root Systems

In this work, we study the concept of the length function and some of its combinatorial properties for the class of extended affine root systems of type A1. We introduce a notion of root basis for these root systems, and using a unique expression of the elements of the Weyl group with respect to a set of generators for the Weyl group, we calculate the length function with respect to a very specific root basis.

scholarly journals Generalized q-Schur algebras and modular representation theory of finite groups with split (BN)-pairs

1999 ◽  
Vol 1999 (511) ◽  
pp. 145-191 ◽  
Author(s):  
Richard Dipper ◽  
Jochen Gruber
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Parabolic Type ◽  
Modular Representation ◽  
Weyl Groups ◽  
Modular Representation Theory ◽  
Schur Algebras ◽  
Schur Algebra ◽  
A Finite Group ◽  
Groups Of Lie Type

Abstract We introduce a generalized version of a q-Schur algebra (of parabolic type) for arbitrary Hecke algebras over extended Weyl groups. We describe how the decomposition matrix of a finite group with split BN-pair, with respect to a non-describing prime, can be partially described by the decomposition matrices of suitably chosen q-Schur algebras. We show that the investigated structures occur naturally in finite groups of Lie type.

scholarly journals Modular Representations of Finite Groups with Unsaturated Split (B,N)-Pairs

1980 ◽  
Vol 32 (3) ◽  
pp. 714-733 ◽  
Author(s):  
N. B. Tinberg
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Prime Number ◽  
Semidirect Product ◽  
Finite Groups ◽  
Modular Representations ◽  
Characteristic P ◽  
Representations Of Finite Groups ◽  
A Finite Group

1. Introduction.Let p be a prime number. A finite group G = (G, B, N, R, U) is called a split(B, N)-pair of characteristic p and rank n if(i) G has a (B, N)-pair (see [3, Definition 2.1, p. B-8]) where H= B ⋂ N and the Weyl group W= N/H is generated by the set R= ﹛ω 1,… , ω n) of “special generators.”(ii) H= ⋂n∈N n-1Bn(iii) There exists a p-subgroup U of G such that B = UH is a semidirect product, and H is abelian with order prime to p.A (B, N)-pair satisfying (ii) is called a saturated (B, N)-pair. We call a finite group G which satisfies (i) and (iii) an unsaturated split (B, N)- pair. (Unsaturated means “not necessarily saturated”.)

scholarly journals On Splitting of the Normalizer of a Maximal Torus in E6(q)

2019 ◽  
Vol 26 (02) ◽  
pp. 329-350
Author(s):  
Alexey Galt ◽  
Alexey Staroletov
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Maximal Torus ◽  
Group Of Lie Type ◽  
Maximal Tori ◽  
A Finite Group ◽  
Simply Connected

Let G be a finite group of Lie type E6 over 𝔽q (adjoint or simply connected) and W be the Weyl group of G. We describe maximal tori T such that T has a complement in its algebraic normalizer N(G, T). It is well known that for each maximal torus T of G there exists an element w ∊ W such that N(G, T )/T ≃ CW(w). When T does not have a complement isomorphic to CW(w), we show that w has a lift in N(G, T) of the same order.

ON THE FIXED-POINT SETS OF TORUS ACTIONS ON FLAG MANIFOLDS

2002 ◽  
Vol 01 (03) ◽  
pp. 255-265 ◽  
Author(s):  
BARBARA A. SHIPMAN
Keyword(s):  
Fixed Point ◽  
Toda Lattice ◽  
Flag Manifold ◽  
Weyl Groups ◽  
Flag Manifolds ◽  
Unipotent Group ◽  
Point Sets ◽  
Torus Actions ◽  
Maximal Tori ◽  
Fixed Point Sets

This paper takes a detailed look at a subject that occurs in various contexts in mathematics, the fixed-point sets of torus actions on flag manifolds, and considers it from the (perhaps nontraditional) perspective of moment maps and length functions on Weyl groups. The approach comes from earlier work of the author where it is shown that certain singular flows in the Hamiltonian system known as the Toda lattice generate the action of a group A on a flag manifold, where A is a direct product of a non-maximal torus and unipotent group. As a first step in understanding the orbits of A in connection with the Toda lattice, this paper seeks to understand the fixed points of the non-maximal tori in this setting.

Weyl ǵroups and finite Chevalley ǵroups

1970 ◽  
Vol 67 (2) ◽  
pp. 269-276 ◽  
Author(s):  
R. W. Carter
Keyword(s):  
Finite Group ◽  
Finite Field ◽  
Lie Algebra ◽  
Weyl Groups ◽  
Chevalley Groups ◽  
Simple Lie Algebra ◽  
Positive Roots ◽  
Image Position ◽  
A Finite Group ◽  
Fundamental Paper

In his fundamental paper (1) Chevalley showed how to associate with each complex simple Lie algebra L and each field K a group G = L(K) which is (in all but four exceptional cases) simple. If K is a finite field GF(q), G is a finite group of orderwhere l is the rank of L, m is the number of positive roots of L and d is a certain integer determined by L and K. The integers m1, m2,…,m1 are determined by L only and satisfy the condition

scholarly journals A Gelfand Model for Weyl Groups of Type D2n

ISRN Algebra ◽  
2012 ◽  
Vol 2012 ◽  
pp. 1-16
Author(s):  
José O. Araujo ◽  
Luis C. Maiarú ◽  
Mauro Natale
Keyword(s):  
Finite Group ◽  
Weyl Algebra ◽  
Polynomial Model ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
Complex Numbers ◽  
Irreducible Factor ◽  
Direct Sum ◽  
Finite Dimensional ◽  
A Finite Group

A Gelfand model for a finite group G is a complex representation of G, which is isomorphic to the direct sum of all irreducible representations of G. When G is isomorphic to a subgroup of GLn(ℂ), where ℂ is the field of complex numbers, it has been proved that each G-module over ℂ is isomorphic to a G-submodule in the polynomial ring ℂ[x1,…,xn], and taking the space of zeros of certain G-invariant operators in the Weyl algebra, a finite-dimensional G-space 𝒩G in ℂ[x1,…,xn] can be obtained, which contains all the simple G-modules over ℂ. This type of representation has been named polynomial model. It has been proved that when G is a Coxeter group, the polynomial model is a Gelfand model for G if, and only if, G has not an irreducible factor of type D2n, E7, or E8. This paper presents a model of Gelfand for a Weyl group of type D2n whose construction is based on the same principles as the polynomial model.

ON THE COXETER COMPLEX AND ALVIS–CURTIS DUALITY FOR PRINCIPAL ℓ-BLOCKS OF GLn(q)

2005 ◽  
Vol 04 (03) ◽  
pp. 225-229 ◽  
Author(s):  
MARKUS LINCKELMANN ◽  
SIBYLLE SCHROLL
Keyword(s):  
Finite Group ◽  
Weyl Group ◽  
Linear Groups ◽  
Homotopy Equivalence ◽  
Derived Equivalence ◽  
Coxeter Complex ◽  
General Linear Groups ◽  
Bounded Complex ◽  
A Finite Group ◽  
Special Case

M. Cabanes and J. Rickard showed in [3] that the Alvis–Curtis character duality of a finite group of Lie type is induced in non defining characteristic ℓ by a derived equivalence given by tensoring with a bounded complex X, and they further conjecture that this derived equivalence should actually be a homotopy equivalence. Following a suggestion of R. Kessar, we show here for the special case of principal blocks of general linear groups with abelian Sylow-ℓ-subgroups that this is true, by an explicit verification relating the complex X to the Coxeter complex of the corresponding Weyl group.

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