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 Number of Classes of a Finite Group Invariant for Certain Substitutions

1974 ◽  
Vol 26 (5) ◽  
pp. 1090-1097 ◽  
Author(s):  
A. J. van Zanten ◽  
E. de Vries
Keyword(s):  
Finite Group ◽  
Irreducible Representation ◽  
Symmetric Group ◽  
General Linear Group ◽  
Irreducible Representations ◽  
Complex Numbers ◽  
Group Invariant ◽  
A Finite Group ◽  
Number Of Classes ◽  
Special Case

In this paper we consider representations of groups over the field of the complex numbers.The nth-Kronecker power σ⊗n of an irreducible representation σ of a group can be decomposed into the constituents of definite symmetry with respect to the symmetric group Sn. In the special case of the general linear group GL(N) in N dimensions the decomposition of the defining representation at once provides irreducible representations of GL(N) [9; 10; 11].

scholarly journals A decomposition of equivariant K-theory in twisted equivariant K-theories

2017 ◽  
Vol 28 (02) ◽  
pp. 1750016 ◽  
Author(s):  
José Manuel Gómez ◽  
Bernardo Uribe
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Irreducible Representation ◽  
Isomorphism Class ◽  
Irreducible Representations ◽  
Direct Sum ◽  
K Theory ◽  
Group 2 ◽  
A Finite Group

For [Formula: see text] a finite group and [Formula: see text] a [Formula: see text]-space on which a normal subgroup [Formula: see text] acts trivially, we show that the [Formula: see text]-equivariant [Formula: see text]-theory of [Formula: see text] decomposes as a direct sum of twisted equivariant [Formula: see text]-theories of [Formula: see text] parametrized by the orbits of the conjugation action of [Formula: see text] on the irreducible representations of [Formula: see text]. The twists are group 2-cocycles which encode the obstruction of lifting an irreducible representation of [Formula: see text] to the subgroup of [Formula: see text] which fixes the isomorphism class of the irreducible representation.

Induced Modules of Semisimple Hopf Algebras

2007 ◽  
Vol 14 (04) ◽  
pp. 571-584 ◽  
Author(s):  
Jun Hu ◽  
Yinhuo Zhang
Keyword(s):  
Finite Group ◽  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Direct Sum ◽  
Drinfel’D Double ◽  
Finite Dimensional ◽  
Induced Module ◽  
A Finite Group

Let K be a field. Let H be a finite-dimensional K-Hopf algebra and D(H) be the Drinfel'd double of H. In this paper, we study Radford's induced module Hβ, where β is a group-like element in H∗. Using the commuting pair established in [7], we obtain an analogue of the class equation for [Formula: see text] when H is semisimple and cosemisimple. In case H is a finite group algebra or a factorizable semisimple cosemisimple Hopf algebra, we give an explicit decomposition of each Hβ into a direct sum of simple D(H)-modules.

scholarly journals Certain representation algebras

1965 ◽  
Vol 5 (1) ◽  
pp. 83-99 ◽  
Author(s):  
S. B. Conlon
Keyword(s):  
Finite Group ◽  
Tensor Product ◽  
Alternating Group ◽  
Complex Numbers ◽  
Direct Sum ◽  
Characteristic 2 ◽  
A Finite Group ◽  
Inequivalent Representations ◽  
Representation Algebra

Let Λ be the set of inequivalent representations of a finite group over a field . Λ is made the basis of an algebra over the complex numbers , called the representation algebra, in which multiplication corresponds to the tensor product of representations and addition to direct sum. Green [5] has shown that if char (the non-modular case) or if is cyclic, then is semi-simple, i.e. is a direct sum of copies of . Here we consider two modular, non-cyclic cases, viz, where is or 4 (alternating group) and is of characteristic 2.

L'application des representations au calcul explicite sur certaines chaînes de Markov finies

1984 ◽  
Vol 16 (3) ◽  
pp. 656-666 ◽  
Author(s):  
Bernard Ycart
Keyword(s):  
Finite Group ◽  
Random Walk ◽  
Irreducible Representations ◽  
Permutation Groups ◽  
A Finite Group

We give here concrete formulas relating the transition generatrix functions of any random walk on a finite group to the irreducible representations of this group. Some examples of such explicit calculations for the permutation groups A4, S4, and A5 are included.

scholarly journals A note on the fixed subring of an FPF ring

1989 ◽  
Vol 40 (1) ◽  
pp. 109-111 ◽  
Author(s):  
John Clark
Keyword(s):  
Finite Group ◽  
Associative Ring ◽  
Finitely Generated ◽  
Group Of Automorphisms ◽  
Direct Sum ◽  
Epimorphic Image ◽  
A Finite Group

An associative ring R with identity is called a left (right) FPF ring if given any finitely generated faithful left (right) R-module A and any left (right) R-module M then M is the epimorphic image of a direct sum of copies of A. Faith and Page have asked if the subring of elements fixed by a finite group of automorphisms of an FPF ring need also be FPF. Here we present examples showing the answer to be negative in general.

scholarly journals A Krull-Schmdit type theorem for coherent sheaves

2014 ◽  
Vol 22 (2) ◽  
pp. 51-56
Author(s):  
A. S. Argáez
Keyword(s):  
Finite Group ◽  
Projective Variety ◽  
Type Theorem ◽  
Coherent Sheaf ◽  
Irreducible Representations ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Coherent Sheaves ◽  
A Finite Group ◽  
Natural Decomposition

AbstractLet X be projective variety over an algebraically closed field k and G be a finite group with g.c.d.(char(k), |G|) = 1. We prove that any representations of G on a coherent sheaf, ρ : G → End(ℰ), has a natural decomposition ℰ ≃ ⊕ V ⊗k ℱV, where G acts trivially on ℱV and the sum run over all irreducible representations of G over k.

On Extending Projectives of Finite Group-Graded Algebras

1991 ◽  
Vol 34 (2) ◽  
pp. 224-228
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Group Representation ◽  
Sufficient Conditions ◽  
Projective Cover ◽  
Graded Algebras ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
A Finite Group ◽  
Necessary And Sufficient

AbstractLet G be a finite group, let k be a field and let R be a finite dimensional fully G-graded k-algebra. Also let L be a completely reducible R-module and let P be a projective cover of R. We give necessary and sufficient conditions for P|R1 to be a projective cover of L|R1 in Mod (R1). In particular, this happens if and only if L is R1-projective. Some consequences in finite group representation theory are deduced.

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.

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