HOMOLOGY DECOMPOSITIONS FOR CLASSIFYING SPACES OF FINITE GROUPS ASSOCIATED TO MODULAR REPRESENTATIONS

2001 ◽  
Vol 64 (2) ◽  
pp. 472-488 ◽  
Author(s):  
D. NOTBOHM
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Homotopy Theory ◽  
Modular Representation ◽  
Modular Representations ◽  
Classifying Spaces ◽  
Modular Representation Theory ◽  
Classifying Space ◽  
Homotopy Colimit ◽  
A Finite Group

For a prime p, a homology decomposition of the classifying space BG of a finite group G consist of a functor F : D → spaces from a small category into the category of spaces and a map hocolim F → BG from the homotopy colimit to BG that induces an isomorphism in mod-p homology. Associated to a modular representation G → Gl(n; [ ]p), a family of subgroups is constructed that is closed under conjugation, which gives rise to three different homology decompositions, the so-called subgroup, centralizer and normalizer decompositions. For an action of G on an [ ]p-vector space V, this collection consists of all subgroups of G with nontrivial p-Sylow subgroup which fix nontrivial (proper) subspaces of V pointwise. These decomposition formulas connect the modular representation theory of G with the homotopy theory of BG.

Weights for Covering Groups of Symmetric and Alternating Groups, р ≠ 2

1991 ◽  
Vol 43 (4) ◽  
pp. 792-813 ◽  
Author(s):  
G. O. Michler ◽  
J. B. Olsson
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Conjugacy Class ◽  
Finite Groups ◽  
Irreducible Character ◽  
Modular Representation ◽  
Modular Representation Theory ◽  
A Finite Group ◽  
Covering Groups ◽  
Fundamental Paper

In his fundamental paper [1] J. L. Alperin introduced the idea of a weight in modular representation theory of finite groups G. Let p be a prime. A p-subgroup R is called a radical subgroup of G if R = Op(NG(R)). An irreducible character φ of NG(R) is called a weight character if φ is trivial on R and belongs to a p-block of defect zero of NG(R)/R. The G-conjugacy class of the pair (R, φ) is a weight of G. Let b be the p-block of NG(R) containing φ, and let B be p-block of G. A weight (R, φ) is a B-weight for the block B of G if B = bG, which means that B and b correspond under the Brauer homomorphism. Alperin's conjecture on weights asserts that the number l*(B) of B-weights of a p-block B of a finite group G equals the number l(B) of modular characters of B.

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.

Stable decompositions of classifying spaces of finite abelianp-groups

1988 ◽  
Vol 103 (3) ◽  
pp. 427-449 ◽  
Author(s):  
John C. Harris ◽  
Nicholas J. Kuhn
Keyword(s):  
Finite Group ◽  
Cohomology Theory ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Image Position ◽  
A Finite Group

LetBGbe the classifying space of a finite groupG. Consider the problem of finding astabledecompositionintoindecomposablewedge summands. Such a decomposition naturally splitsE*(BG), whereE* is any cohomology theory.

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 Brauer’s height 0 conjecture

1988 ◽  
Vol 109 ◽  
pp. 109-116 ◽  
Author(s):  
T.R. Berger ◽  
R. Knörr
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Groups ◽  
Modular Representation ◽  
Defect Group ◽  
Modular Representation Theory

R. Brauer not only laid the foundations of modular representation theory of finite groups, he also raised a number of questions and made conjectures (see [1], [2] for instance) which since then have attracted the interest of many people working in the field and continue to guide the research efforts to a good extent. One of these is known as the “Height zero conjecture”. It may be stated as follows: CONJECTURE. Let B be a p-block of the finite group G. All irreducible ordinary characters of G belonging to B are of height 0 if and only if a defect group of B is abelian.

Analogs of the Green and Puig correspondences for twisted group algebras

2016 ◽  
Vol 19 (1) ◽  
pp. 1-24
Author(s):  
Morton E. Harris
Keyword(s):  
Finite Group ◽  
Representation Theory ◽  
Finite Groups ◽  
Modular Representation ◽  
Group Algebras ◽  
Modular Representation Theory ◽  
Twisted Group ◽  
Standard Derivation ◽  
Green Correspondence

AbstractIn the modular representation theory of finite groups, we show that the standard derivation of the Green correspondence lifts to a derivation of a Green correspondence for twisted group algebras (Theorem 1.3). Then, from these results we derive a lift of the Puig correspondences for twisted group algebras (Theorem 1.6).Clearly twisted group algebras arise naturally in finite group modular representation theory. We conclude with some suggestions for applications in this mathematical area.

scholarly journals On the kernels of representations of finite groups

1981 ◽  
Vol 22 (2) ◽  
pp. 151-154 ◽  
Author(s):  
Shigeo Koshitani
Keyword(s):  
Finite Group ◽  
Prime Number ◽  
Finite Groups ◽  
Solvable Groups ◽  
Modular Representation ◽  
Complex Representation ◽  
Representations Of Finite Groups ◽  
A Finite Group

Let G be a finite group and p a prime number. About five years ago I. M. Isaacs and S. D. Smith [5] gave several character-theoretic characterizations of finite p-solvable groups with p-length 1. Indeed, they proved that if P is a Sylow p-subgroup of G then the next four conditions (l)–(4) are equivalent:(1) G is p-solvable of p-length 1.(2) Every irreducible complex representation in the principal p-block of G restricts irreducibly to NG(P).(3) Every irreducible complex representation of degree prime to p in the principal p-block of G restricts irreducibly to NG(P).(4) Every irreducible modular representation in the principal p-block of G restricts irreducibly to NG(P).

scholarly journals Commuting elements, simplicial spaces and filtrations of classifying spaces

2011 ◽  
Vol 152 (1) ◽  
pp. 91-114 ◽  
Author(s):  
ALEJANDRO ADEM ◽  
FREDERICK R. COHEN ◽  
ENRIQUE TORRES GIESE
Keyword(s):  
Finite Group ◽  
Central Element ◽  
Simple Groups ◽  
Central Series ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Odd Order ◽  
A Finite Group ◽  
Counting Formula

AbstractLet G denote a topological group. In this paper the descending central series of free groups are used to construct simplicial spaces of homomorphisms with geometric realizations B(q, G) that provide a filtration of the classifying space BG. In particular this setting gives rise to a single space constructed out of all the spaces of ordered commuting n–tuples of elements in G. Basic properties of these constructions are discussed, including the homotopy type and cohomology when the group G is either a finite group or a compact connected Lie group. For a finite group the construction gives rise to a covering space with monodromy related to a delicate result in group theory equivalent to the odd-order theorem of Feit–Thompson. The techniques here also yield a counting formula for the cardinality of Hom(π, G) where π is any descending central series quotient of a finitely generated free group. Another application is the determination of the structure of the spaces B(2, G) obtained from commuting n-tuples in G for finite groups such that the centralizer of every non–central element is abelian (known as transitively commutative groups), which played a key role in work by Suzuki on the structure of finite simple groups.

scholarly journals Massey products in the homology of the loop space of a p-completed classifying space: finite groups with cyclic Sylow p-subgroups

2021 ◽  
pp. 1-8
Author(s):  
Dave Benson ◽  
John Greenlees
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Loop Space ◽  
Classifying Space ◽  
Massey Products ◽  
A Finite Group

Abstract Let $G$ be a finite group with cyclic Sylow $p$ -subgroups, and let $k$ be a field of characteristic $p$ . Then $H^{*}(BG;k)$ and $H_*(\Omega BG{{}^{{}^{\wedge }}_p};k)$ are $A_\infty$ algebras whose structure we determine up to quasi-isomorphism.

scholarly journals RETRACTIVE TRANSFERS AND p-LOCAL FINITE GROUPS

2008 ◽  
Vol 51 (2) ◽  
pp. 465-487 ◽  
Author(s):  
Kári Ragnarsson
Keyword(s):  
Finite Group ◽  
Fundamental Group ◽  
Finite Groups ◽  
Stable Homotopy ◽  
Classifying Spaces ◽  
Classifying Space ◽  
Frobenius Reciprocity ◽  
Equivalent Theory ◽  
Natural Inclusion ◽  
Transfer Properties

AbstractIn this paper we explore the possibility of defining $p$-local finite groups in terms of transfer properties of their classifying spaces. More precisely, we consider the question, posed by Haynes Miller, of whether an equivalent theory can be obtained by studying triples $(f,t,X)$, where $X$ is a $p$-complete, nilpotent space with a finite fundamental group, $f:BS\to X$ is a map from the classifying space of a finite $p$-group, and $t$ is a stable retraction of $f$ satisfying Frobenius reciprocity at the level of stable homotopy. We refer to $t$ as a retractive transfer of $f$ and to $(f,t,X)$ as a retractive transfer triple over $S$.In the case where $S$ is elementary abelian, we answer this question in the affirmative by showing that a retractive transfer triple $(f,t,X)$ over $S$ does indeed induce a $p$-local finite group over $S$ with $X$ as its classifying space.Using previous results obtained by the author, we show that the converse is true for general finite $p$-groups. That is, for a $p$-local finite group $(S,\mathcal{F},\mathcal{L})$, the natural inclusion $\theta:BS\to X$ has a retractive transfer $t$, making $(\theta,t,|\mathcal{L}|^{\wedge}_p)$ a retractive transfer triple over $S$. This also requires a proof, obtained jointly with Ran Levi, that $|\mathcal{L}|^{\wedge}_p$ is a nilpotent space, which is of independent interest.

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