On Perfect Isometries for Tame Blocks

Any 2-block of a finite group G with a quaternion defect groupQ₈ is Morita equivalent to the corresponding block of the centraliserH of the unique involution of Q₈ in G; this answers positivelyan earlier question raised by M. Broué. 2000 MathematicsSubject Classification 20C20.     

Symmetric Groups, Wreath Products, Morita Equivalences, and Broue's Abelian Defect Group Conjecture

It is shown that for any prime p, and any non-negative integerw less than p, there exist p-blocks of symmetric groups of defectw, which are Morita equivalent to the principal p-block of thegroup S_p S_w. Combined with work of J. Rickard, this provesthat Broué's abelian defect group conjecture holds forp-blocks of symmetric groups of defect at most 5.     

Morita Equivalent 3-Blocks of the 3-Dimensional Projective Special Linear Groups

If G is a projective special linear group PSL(3,q) with q 4or 7 (mod 9), then a Sylow 3-subgroup of G is elementary abelianof order 9. We show that the principal 3-blocks of any two suchgroups are Morita equivalent. This result and Okuyama's theoremfor PSL(3,4) prove the Broué conjecture for these blocks.1991 Mathematics Subject Classification: 20C05, 20C20.     

Broué's conjecture for non-principal 3-blocks of finite groups

In representation theory of finite groups, there is a well-known and important conjecture due to M. Broué. He conjectures that, for any prime p, if a p-block A of a finite group G has an abelian defect group D, then A and its Brauer correspondent p-block B of N_G(D) are derived equivalent. We demonstrate in this paper that Broué's conjecture holds for two non-principal 3-blocks A with elementary abelian defect group D of order 9 of the O'Nan simple group and the Higman-Sims simple group. Moreover, we determine these two non-principal block algebras over a splitting field of characteristic 3 up to Morita equivalence.     

Splendid Derived Equivalences for Blocks of Finite Groups

A central issue in finite group modular representation theoryis the relationship between the p-local structure and the p-modularrepresentation theory of a given finite group. In [5], Brouéposes some startling conjectures. For example, he conjecturesthat if e is a p-block of a finite group G with abelian defectgroup D and if f is the Brauer correspondent block of e of thenormalizer, N_G(D), of D then e and f have equivalent derivedcategories over a complete discrete valuation ring with residuefield of characteristic p. Some evidence for this conjecturehas been obtained using an important Morita analog for derivedcategories of Rickard [11]. This result states that the existenceof a tilting complex is a necessary and sufficient conditionfor the equivalence of two derived categories. In [5], Brouéalso defines an equivalence on the character level between p-blockse and f of finite groups G and H that he calls a ‘perfectisometry’ and he demonstrates that it is a consequenceof a derived category equivalence between e and f. In [5], Brouéalso poses a corresponding perfect isometry conjecture betweena p-block e of a finite group G with an abelian defect groupD and its Brauer correspondent p-block f of N_G(D) and presentsseveral examples of this phenomena. Subsequent research hasprovided much more evidence for this character-level conjecture. In many known examples of a perfect isometry between p-blockse, f of finite groups G, H there are also perfect isometriesbetween p-blocks of p-local subgroups corresponding to e andf and these isometries are compatible in a precise sense. In[5], Broué calls such a family of compatible perfectisometries an ‘isotypy’. In [11], Rickard addresses the analogous question of defininga p-locally compatible family of derived equivalences. In thisimportant paper, he defines a ‘splendid tilting complex’for p-blocks e and f of finite groups G and H with a commonp-subgroup P. Then he demonstrates that if X is such a splendidtilting complex, if P is a Sylow p-subgroup of G and H and ifG and H have the same ‘p-local structure’, thenp-local splendid tilting complexes are obtained from X via theBrauer functor and ‘lifting’. Consequently, in thissituation, we obtain an isotypy when e and f are the principalblocks of G and H. Linckelmann [9] and Puig [10] have also obtained important resultsin this area. In this paper, we refine the methods and program of [11] toobtain variants of some of the results of [11] that have widerapplicability. Indeed, suppose that the blocks e and f of Gand H have a common defect group D. Suppose also that X is asplendid tilting complex for e and f and that the p-local structureof (say) H with respect to D is contained in that of G, thenthe Brauer functor, lifting and ‘cutting’ by blockindempotents applied to X yield local block tilting complexesand consequently an isotypy on the character level. Since thep-local structure containment hypothesis is satisfied, for example,when H is a subgroup of G (as is the case in Broué'sconjectures) our results extend the applicability of these ideasand methods.     

On The Profinite Topology on a Free Group

If F is a free abstract group, its profinite topology is thecoarsest topology making F into a topological group, such thatevery group homomorphism from F into a finite group is continuous.It was shown by M. Hall Jr that every finitely generated subgroupof F is closed in that topology. Let H₁, H₂, ..., H_n be finitelygenerated subgroups of F. J.-E. Pin and C. Reutenauer have conjecturedthat the product H₁ H₂ ... H_n is a closed set in the profinitetopology of F; also, they have shown that this conjecture impliesa conjecture of J. Rhodes on finite semigroups. In this paperwe give a positive answer to the conjecture of Pin and Reutenauer.Our method is based on the theory of profinite groups actingon graphs.     

From Klein to Painleve Via Fourier, Laplace and Jimbo

We describe a method for constructing explicit algebraic solutionsto the sixth Painlevé equation, generalising that ofDubrovin and Mazzocco. There are basically two steps. Firstwe explain how to construct finite braid group orbits of triplesof elements of SL₂(C) out of triples of generators of three-dimensionalcomplex reflection groups. (This involves the Fourier–Laplacetransform for certain irregular connections.) Then we adapta result of Jimbo to produce the Painlevé VI solutions.(In particular, this solves a Riemann–Hilbert problemexplicitly.) Each step is illustrated using the complex reflection groupassociated to Klein's simple group of order 168. This leadsto a new algebraic solution with seven branches. We also provethat, unlike the algebraic solutions of Dubrovin and Mazzoccoand Hitchin, this solution is not equivalent to any solutioncoming from a finite subgroup of SL₂(C). The results of this paper also yield a simple proof of a recenttheorem of Inaba, Iwasaki and Saito on the action of Okamoto'saffine D₄ symmetry group as well as the correct connection formulaefor generic Painlevé VI equations. 2000 Mathematics SubjectClassification 34M55, 34M40, 20F55.     

Cohomology Actions and Centralisers in Unitary Reflection Groups

In an earlier work, the second author proved a general formulafor the equivariant Poincaré polynomial of a linear transformationg which normalises a unitary reflection group G, acting on thecohomology of the corresponding hyperplane complement. Thisformula involves a certain function (called a Z-function below)on the centraliser CG(g), which was proved to exist only incertain cases, for example, when g is a reflection, or is G-regular,or when the centraliser is cyclic. In this work we prove theexistence of Z-functions in full generality. Applications includereduction and product formulae for the equivariant Poincarépolynomials. The method is to study the poset L(C_G(g)) of subspaceswhich are fixed points of elements of C_G(g). We show that thisposet has Euler characteristic 1, which is the key propertyrequired for the definition of a Z-function. The fact aboutthe Euler characteristic in turn follows from the ‘join-atom’property of L(C_G(g)), which asserts that if [X₁,..., X_k} isany set of elements of L(C_G(g)) which are maximal (set theoretically)then their setwise intersection lies in L(C_G(g)). 2000 Mathematical Subject Classification:primary 14R20, 55R80; secondary 20C33, 20G40.     

Morita Equivalent Principal 3-Blocks of the Chevalley Groups G2(q)

The principal 3-block of a Chevalley group G₂(q) with q a powerof 2 satisfying q 2 or 5 mod 9 and the principal 3-block ofG₂(2) are Morita equivalent. 2000 Mathematical Subject Classification:20C05, 20C20, 20C33.     

A Conjecture on the Hall Topology for the Free Group

The Hall topology for the free group is the coarsest topologysuch that every group morphism from the free group onto a finitediscrete group is continuous. It was shoen by M.Hall Jr thatevery finitely generated subgroup of the free group is closedfor this topology. We conjecture that if H₁, H₂,...,H_n are finitelygenerated subgroups of the free group, then the product H₁ H₂...H_n is closed. We discuss some consequences of this conjecture.First, it would give a nice and simple algorithm to computethe closure of a given rational subset of the free group. Next,it implies a similar conjecture for the free monoid, which inturn is equivalent to a deep conjecture on finite semigroupsfor the solution of which J. Rhodes has offered $100. We hopethat our new conjecture will shed some light on Rhodes' conjecture.     

Twisted Group Algebras, Normal Subgroups and Derived Equivalences

We study Rickard equivalences between p-blocks of twisted group algebras and their local structure, in connection with Dade's conjectures. We prove that an extended form of Broué's conjecture implies Dade's Inductive Conjecture in the Abelian defect group case; this is a consequence of the fact that Rickard equivalences induced by complexes of graded bimodules preserve the relevant Clifford theoretical invariants. As an application, we show that these conjectures hold for p-extensions of blocks with cyclic defect groups.     

Enumerating Transitive Finite Permutation Groups

Denote by f(n) the number of subgroups of the symmetric groupSym(n) of degree n, and by f_trans(n) the number of its transitivesubgroups. It was conjectured by Pyber [9] that almost all subgroupsof Sym(n) are not transitive, that is, f_trans(n)/f(n) tendsto 0 when n tends to infinity. It is still an open questionwhether or not this conjecture is true. The difficulty comesfrom the fact that, from many points of view, transitivity isnot a really strong restriction on permutation groups, and thereare too many transitive groups [9, Sections 3 and 4]. In thispaper we solve the problem in the particular case of permutationgroups of prime power degree, proving the following result.1991 Mathematics Subject Classification 20B05, 20D60.     

Formal Groups of Building Blocks Completely Defined Over Finite Abelian Extensions of Q

Let C be an elliptic curve defined over Q. We can associatetwo formal groups with C: the formal group (X, Y) determinedby the formal completion of the Néron model of C overZ along the zero section, and the formal group F_L(X, Y) of theL-series attached to l-adic representations on C of the absoluteGalois group of Q. Honda shows that F_L(X, Y) is defined overZ, and it is strongly isomorphic over Z to (X, Y). In this paperwe give a generalization of the result of Honda to buildingblocks over finite abelian extensions of Q. The difficulty isto define new matrix L-series of building blocks. Our generalizationcontains the generalization of Deninger and Nart to abelianvarieties of GL₂-type. It also contains the generalization ofour previous paper to Q-curves over quadratic fields. 2000 MathematicsSubject Classification 11G10 (primary), 11F11 (secondary).     

Embeddings of Sz(32) in E8(5)

We show that the Suzuki group Sz(32) is a subgroup of E₈(5),and so is its automorphism group. Both are unique up to conjugacyin E₈(F) for any field F of characteristic 5, and the automorphismgroup Sz(32):5 is maximal in E₈(5). 1991 Mathematics SubjectClassification 20E28.     

On Tensor Products of Modular Representations of Symmetric Groups

Let F be a field, and let _n be the symmetric group on n letters.In this paper we address the following question: given two irreducibleF_n-modules D₁ and D₂ of dimensions greater than 1, can it happenthat D₁ D₂ is irreducible? The answer is known to be ‘no’if char F = 0 [12] (see also [2] for some generalizations).So we assume from now on that F has positive characteristicp. The following conjecture was made in [4]. CONJECTURE. Let D₁ and D₂ be two irreducible F_n-modules of dimensionsgreater than 1. Then D₁ D₂ is irreducible if and only if p= 2, n = 2 + 4l for some positive integer l; one of the modulescorresponds to the partition (2l + 2, 2l) and the other correspondsto a partition of the form (n – 2j – 1, 2j 1), 0 j < l. Moreover, in the exceptional cases, one has The main result of this paper is the following theorem, whichestablishes a big part of the conjecture. 1991 Mathematics SubjectClassification 20C20, 20C30.     

A Bound on the Presentation Rank of a Finite Group

Let G be a finite group, and let I_G be the augmentation idealof ZG. We denote by d(G) the minimum number of generators forthe group G, and by d(I_G) the minimum number of elements ofI_G needed to generate I_G as a G-module. The connection betweend(G) and d(I_G) is given by the following result due to Roggenkamp]14]: where pr(G) is a non-negative integer, called the presentationrank of G, whose definition comes from the study of relationmodules (see [4] for more details). 1991 Mathematics SubjectClassification 20D20.     

A characterization of finite soluble groups

Let G be a finite soluble group of order m and let w(x₁, ...,x_n) be a group word. Then the probability that w(g₁, ..., g_n)= 1 (where (g₁, ..., g_n) is a random n-tuple in G) is at leastp^–(m–t), where p is the largest prime divisor ofm and t is the number of distinct primes dividing m. This contrastswith the case of a non-soluble group G, for which Abérthas shown that the corresponding probability can take arbitrarilysmall positive values as n .     

Euler Characteristics of Coxeter Groups, PL-Triangulations of Closed Manifolds, and Cohomology of Subgroups of Artin Groups

The motivation for the theory of Euler characteristics of groups,which was introduced by C. T. C. Wall [21], was topology, butit has interesting connections to other branches of mathematicssuch as group theory and number theory. This paper investigatesEuler characteristics of Coxeter groups and their applications.In his paper [20], J.-P. Serre obtained several fundamentalresults concerning the Euler characteristics of Coxeter groups.In particular, he obtained a recursive formula for the Eulercharacteristic of a Coxeter group, as well as its relation tothe Poincaré series (see 3). Later, I. M. Chiswell obtainedin [10] a formula expressing the Euler characteristic of a Coxetergroup in terms of orders of finite parabolic subgroups (Theorem1). These formulae enable us to compute Euler characteristicsof arbitrary Coxeter groups. On the other hand, the Euler characteristics of Coxeter groupsW happen to be intimately related to their associated complexesF_W, which are defined by means of the posets of nontrivial parabolicsubgroups of finite order (see 2.1 for the precise definition).In particular, it follows from the recent result of M. W. Davis[13] that if F_W is a product of a simplex and a generalizedhomology 2n-sphere, then the Euler characteristic of W is zero(Corollary 3.1). The first objective of this paper is to generalizethe previously mentioned result to the case when F_W is a PL-triangulationof a closed 2n-manifold which is not necessarily a homology2n-sphere. In other words (as given below in Theorem 3), ifW is a Coxeter group such that F_W is a PL-triangulation of aclosed 2n-manifold, then the Euler characteristic of W is equalto 1–(F_W)/2.     

Counterexamples to Tischler's Strong Form of Smale's Mean Value Conjecture

Smale's mean value conjecture asserts that for every polynomial P of degree d satisfying P(0)=0,where K = (d–1)/d and the minimum is taken over all criticalpoints of P. A stronger conjecture due to Tischler assertsthat with . Tischler's conjecture is known to be true: (i) for local perturbations of the extremumP₀(z)=z^d – dz, and (ii) for all polynomials of degreed 4. In this paper, Tischler's conjecture is verified for alllocal perturbations of the extremum P₁(z)=(z – 1)^d –(–1)^d, but counterexamples to the conjecture are givenin each degree d 5. In addition, estimates for certain weightedL¹- and L²-averages of the quantities are established, which lead to the best currentlyknown value for K₁ in the case d=5. 2000 Mathematics SubjectClassification 30C15.     

Smoothness Properties of the Unit Ball in a JB*-Triple

An element a of norm one in a JB^*-triple A is said to be smoothif there exists a unique element x in the unit ball A₁^* of thedual A^* of A at which a attains its norm, and is said to beFréchet-smooth if, in addition, any sequence (x_n) ofelements in A₁^* for which (x_n(a)) converges to one necessarilyconverges in norm to x. The sequence (a²ⁿ⁺¹) of odd powers ofa converges in the weak^*-topology to a tripotent u(a) in theJBW^*-envelope A^** of A. It is shown that a is smooth if andonly if u(a) is a minimal tripotent in A^** and a is Fréchet-smoothif and only if, in addition, u(a) lies in A.