Alperin's Conjecture Research Articles

Parameterization of Irreducible Characters for p-Solvable Groups

The weights for a finite group G with respect to a prime number p were introduced by Jon Alperin, in order to formulate his celebrated conjecture. In 1992, Everett Dade formulated a refinement of Alperin's conjecture involving ordinary irreducible characters — with their defect — and, in 2000, Geoffrey Robinson proved that the new conjecture holds for p-solvable groups. But this refinement is formulated in terms of a vanishing alternating sum, without giving any possible refinement for the weights. In this note we show that, in the case of the p-solvable finite groups, the method developed in a previous paper can be suitably refined to provide, up to the choice of a polarization ω, a natural bijection — namely compatible with the action of the group of outer automorphisms of G — between the sets of absolutely irreducible characters of G and of G-conjugacy classes of suitable inductive weights, preserving blocks and defects.

On the reduction of Alperin's Conjecture to the quasi-simple groups

We show that the refinement of Alperin's Conjecture proposed in L. Puig (2009) [10, Chap. 16] can be proved by checking that this refinement holds on any central k ⁎ -extension of a finite group H containing a normal simple group S with trivial centralizer in H and p ′ -cyclic quotient H / S . This paper improves our result in L. Puig (2009) [10, Theorem 16.45] and repairs some bad arguments there.

On finite simple groups ofp-local rank two

G. Robinson introduced the group invariant known as the p-local rank to study Dade's conjecture and Alperin's conjecture. It is known that, for a finite p-solvable group with trivial maximal normal p-subgroup, the p-local rank is greater than or equal to the p-rank. Along those lines, we study the p-local rank of finite simple groups, giving a group-theoretic characterization of finite simple groups having p-local rank two. These results are also necessary for the investigation of such conjectures for finite groups of p-local rank two.

Some refinements of Dade's projective conjecture

In this paper a further refinement of Dade's projective conjecture, due to Boltje, is presented. This new statement includes ideas first published by Isaacs and Navarro as well as the recent contractibility version of Alperin's conjecture introduced by Boltje. Leaning heavily on the work of Robinson, weaker forms of the conjecture are proved in the case of p-solvable groups.

Cohomology of Finite Groups: Interactions and Applications

The aim of this meeting was to bring together people from different fields in which the cohomology of finite groups is an important tool. Several such meetings have taken place in the past (for example at Oberwolfach in 2000) and they have contributed substantially to the development of interactions between such fields, in particular between commutative algebra, homological algebra, homotopy theory, modular representation theory and transformation groups. \smallskip The meeting was attended by 54 participants from about a dozen countries. There were 24 talks of various length. Several of them were directly concerned with the cohomology of finite groups, but beyond that there were talks in which the relation of the cohomology of finite groups with other topics was emphasized: among them talks on the cohomology of infinite groups, of quantum groups and of local rings, on functor cohomology, on transformation groups and p -compact groups, on homological algebra and triangulated categories, on endotrivial modules in modular representation theory and on the Alperin conjecture as well as on K -theory, on invariant theory, on commutative algebra and relations to stable homotopy theory. \smallskip The schedule allowed for enough time for extensive and lively interactions between the participants. Besides the traditional and popular Wednesday afternoon hike (during a week with an exceptionally nice weather) an excellent concert was organized on Thursday evening by some of the participants of the conference. As always the very pleasant and stimulating atmosphere at the institute contributed to make this a very successful meeting.

Alperin's Conjecture and the Subgroup Structure of a Finite Group

Alperin's Conjecture and the Subgroup Structure of a Finite Group

Local representation theory and möbius inversion

Various representation-theoretic parameters of a finite group are shown to satisfy formulas similar to formulas appearing in reformulations by Kölshammer, Robinson, and Thévenaz of Alperin's Conjecture. The conjecture itself is reformulated again, now as a statement not mentioning characters or conjugacy classes.

On Alperin's Conjecture and Certain Subgroup Complexes

We prove a new formula about local control of the number of p-regular conjugacyclasses of a finite group. We then relate the results to Alperin’s weight conjecture to obtain newresults describing the number of simple modules for a finite group in terms of weights of solvablesubgroups. Finally, we use the results to obtain new formulations of Alperin’s weight conjecture,and to obtain restrictions on the structure of a minimal counterexample.

Local Structure, Vertices and Alperin's Conjecture

Local Structure, Vertices and Alperin's Conjecture

MÖBIUS INVERSION AND THE LEFSCHETZ INVARIANTS OF SOME p-SUBGROUP COMPLEXES

ABSTRACT: By generalising Möbius inversion to finite acyclic directed multigraphs, we show that the coefficients for the reduced Lefschetz invariant of a suitable G-poset of p-subgroups are, essentially, values of a Möbius function. This allows us to present the Knörr–Robinson reformulation of Alperin's conjecture as a case of Möbius inversion.

MÖBIUS INVERSION AND THE LEFSCHETZ INVARIANTS OF SOME p-SUBGROUP COMPLEXES

ABSTRACT: By generalising Möbius inversion to finite acyclic directed multigraphs, we show that the coefficients for the reduced Lefschetz invariant of a suitable G-poset of p-subgroups are, essentially, values of a Möbius function. This allows us to present the Knörr–Robinson reformulation of Alperin's conjecture as a case of Möbius inversion.

The defect groups of a clique, p-solvable groups, and Alperin's conjecture.

The defect groups of a clique, p-solvable groups, and Alperin's conjecture.

Alperin's Conjecture, Numbers of Characters, and Euler Characteristics of Quotients of p -Group Complexes

Alperin's Conjecture, Numbers of Characters, and Euler Characteristics of Quotients of <i>p</i> -Group Complexes

The Alperin Conjecture for Wreath-Products of the Form GwrS n

The Alperin Conjecture for Wreath-Products of the Form GwrS n

Equivariant K-theory and Alperin's conjecture

We show that Alperin's conjecture in the modular representation theory of a finite group G is equivalent to a conjecture about the equivariant K-theory of the simplicial complex of p-subgroups of G

On conjectures of Olsson, Brauer, and Alperin

Let G be a finite group, p a prime number, B a p-block of the group G, k(B) the number of irreducible complex characters of R belonging to B, k0(B) the number of irreducible characters of height zero in B, and let D be the defect group of B. This article considers the relationship between Brauer's conjecture (k(B) ≤ ¦D¦), Olsson's conjecture (k0(B) ≤¦D/D'¦), and Alperin's conjecture (k0(B) = k0(~B), where ~B is a p-block NG(D) such that ~BG = B). In particular, Olsson's conjecture is proved for p-blocks for those p-solvable groups G for which a Hall p′-subgroup of the group NG(D) is either supersolvable or has odd order.

Locally Determined Functions and Alperin's Conjecture

Keywords: Brown complex ; posets of subgroups ; finite groups ; conjugacy classes ; p-locally determined functions ; p-local subgroups ; Alperin's conjecture ; absolutely irreducible representations Reference CTG-ARTICLE-1992-005doi:10.1112/jlms/s2-45.3.446 Record created on 2008-12-16, modified on 2017-05-12

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

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.

Character correspondences in finite general linear, unitary and symmetric groups

Let r > 0 be a prime integer, and let G be a finite general linear group GL(n, q), a unitary group U(n, q) with q = qg, or a symmetric group S(n) of degree n. The partition of the irreducible (ordinary) characters of S(n) into r-blocks of S(n) was given by Brauer's and Robinson's solution of Nakayama's conjecture, see [7], p. 245. In their fundamental paper [5] Fong and Srinivasan have recently classified the r-blocks of GL(n, q) and U(n, q) for all primes r > 2 with (r, q)= 1. Using these classifications of the r-blocks of G we show in this article that there is a natural one-to-one correspondence ~b between the irreducible characters of height zero of an r-block B of G with defect group R and the irreducible characters of height zero of the Brauer correspondent b of B in N=NG(R ) (Theorem (4.10)). In particular, ko(B)=ko(b), where ko(B ) denotes the number of all irreducible characters )~ of B with height h t z = 0 . Therefore Alperin's conjecture on the numbers of irreducible characters of height zero is verified for all general linear, unitary and symmetric groups. If G=S(n), then Theorem(4.10) also holds for r = 2. In order to establish the character correspondence ~, three other correspondences are studied, the product of which is ~b. In Sect. 1 we construct for every r-block B of G with defect group R a subgroup G of G with a Sylow r-subgroup /~-~R such that there is a natural height preserving one-to-one correspondence T between the set Irr(B) of all irreducible characters of B and the set Irr(/~0) of all irreducible characters of the principal r-block/~o of d (Reduction Theorems (1.9) and (1.10)). The map ~ respects the geometric conjugacy classes of characters. If /~0 denotes the principal r-block of N=Nd(/~), and if b is the Brauer correspondent of B in N=N6(R), then by Theorems (3.8) and (3.10) the block ideals/~o and b are Morita equivalent, have the same decomposition numbers, and there is a natural height preserving one-to-one correspondence a between Irr(bo) and Irr(b).