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.

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.

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 Cohomology Theory for Non-Normal Subgroups and Non-Normal Fields

1954 ◽  
Vol 2 (2) ◽  
pp. 66-76 ◽  
Author(s):  
Iain T. Adamson
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Cohomology Group ◽  
Cohomology Theory ◽  
Normal Subgroups ◽  
Cohomology Groups ◽  
Coset Space ◽  
Image Position ◽  
A Finite Group

Let G be a finite group, H an arbitrary subgroup (i.e., not necessarily normal); we decompose G as a union of left cosets modulo H:choosing fixed coset representatives v. In this paper we construct a “coset space complex” and assign cohomology groups; Hr([G: H], A), to it for all coefficient modules A and all dimensions, -∞<r<∞. We show that ifis an exact sequence of coefficient modules such that H1U, A')= 0 for all subgroups U of H, then a cohomology group sequencemay be defined and is exact for -∞<r<∞. We also provide a link between the cohomology groups Hr([G: H], A) and the cohomology groups of G and H; namely, we prove that if Hv(U, A)= 0 for all subgroups U of H and for v = 1, 2, …, n–1, then the sequenceis exact, where the homomorphisms of the sequence are those induced by injection, inflation and restriction respectively.

scholarly journals Integral representations with trivial first cohomology groups

1982 ◽  
Vol 85 ◽  
pp. 231-240 ◽  
Author(s):  
Shizuo Endo ◽  
Takehiko Miyata
Keyword(s):  
Finite Group ◽  
Equivalence Relation ◽  
Integral Representations ◽  
Cohomology Groups ◽  
Finitely Generated ◽  
Abelian Semigroup ◽  
A Finite Group

Let Π be a finite group and denote by MΠ the class of finitely generated Z-free ZΠ-modules. In [2] we defined a certain equivalence relation on MΠ and constructed the abelian semigroup T(Π), which was studied in [3] (see [1] and [5], too).

scholarly journals Homotopy Formulas for Cyclic Groups Acting on Rings

2008 ◽  
Vol 51 (1) ◽  
pp. 81-85
Author(s):  
Christian Kassel
Keyword(s):  
Finite Group ◽  
Cyclic Group ◽  
Cohomology Groups ◽  
Joint Work ◽  
Cyclic Groups ◽  
A Finite Group

AbstractThe positive cohomology groups of a finite group acting on a ring vanish when the ring has a norm one element. In this note we give explicit homotopies on the level of cochains when the group is cyclic, which allows us to express any cocycle of a cyclic group as the coboundary of an explicit cochain. The formulas in this note are closely related to the effective problems considered in previous joint work with Eli Aljadeff.

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

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 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.

Characteristic classes for permutation representations

1981 ◽  
Vol 90 (2) ◽  
pp. 265-272 ◽  
Author(s):  
G. B. Segal ◽  
C. T. Stretch
Keyword(s):  
Finite Group ◽  
Characteristic Classes ◽  
Cohomology Groups ◽  
Trivial Representation ◽  
Real Representation ◽  
Finite Dimensional ◽  
Whitney Class ◽  
Image Position ◽  
A Finite Group

To a finite-dimensional real representation V of a finite group G there are associated its Stiefel–Whitney classes wk (V) (k = 1, 2, 3, …) in the cohomology groups Hk(G; ). ( is the field with two elements.) The total Stiefel-Whitney classin the ring H*(G; is natural with respect to G in the obvious sense, and, in addition,(a) exponential, i.e. w(V ⊕ W) = w(V).w(W),and(b) stable, i.e. w(V) = 1 when F is a trivial representation.

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