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.     

Burns' Equivariant Tamagawa Invariant T{Omega}loc(N/Q,1) for some Quaternion Fields

Inspired by the work of Bloch and Kato in [2], David Burns constructedseveral ‘equivariant Tamagawa invariants’ associatedto motives of number fields. These invariants lie in relativeK-groups of group-rings of Galois groups, and in [3] Burns gaveseveral conjectures (see Conjecture 3.1) about their values.In this paper I shall verify Burns' conjecture concerning theinvariant T^loc(N/Q,1) for some families of quaternion extensionsN/Q. Using the results of [9] I intend in a subsequent paperto verify Burns' conjecture for those families of quaternionfields which are not covered here.     

Modular Form Congruences and Selmer Groups

Motivated by Cremona and Mazur's notion of visibility of elementsin Shafarevich–Tate groups [6, 27], there have been anumber of recent works which test its compatibility with theBirch and Swinnerton–Dyer conjecture and the Bloch–Katoconjecture. These conjectures provide formulas for the ordersof Shafarevich–Tate groups in terms of values of L-functions.For example, one may see recent work of Agashe, Dummigan, Steinand Watkins [1, 2, 10, 11]. In their examples, they find thatthe presence of visible elements agrees with the expected divisibilityproperties of the relevant L-values.     

Cores of Hyperbolic 3-Manifolds and Limits of Kleinian Groups II

Troels Jørgensen conjectured that the algebraic and geometriclimits of an algebraically convergent sequence of isomorphicKleinian groups agree if there are no new parabolics in thealgebraic limit. We prove that this conjecture holds in ‘most’cases. In particular, we show that it holds when the domainof discontinuity of the algebraic limit of such a sequence isnon-empty (see Theorem 3.1). We further show, with the sameassumptions, that the limit sets of the groups in the sequenceconverge to the limit set of the algebraic limit. As a corollary,we verify the conjecture for finitely generated Kleinian groupswhich are not (non-trivial) free products of surface groupsand infinite cyclic groups (see Corollary 3.3). These resultsare extensions of similar results for purely loxodromic groupswhich can be found in [4]. Thurston [32] previously establishedthese results in the case when the Kleinian groups are freelyindecomposable (see also Ohshika [24, 25, 27]). Using differenttechniques from ours, Ohshika [26] has proven versions of theseresults for purely loxodromic function groups.     

p-Divided Lie Rings and p-Groups

In the Kourovka Notebook [7] Khukhro posed a conjecture on thestructure of finite p-groups admitting an automorphism of orderp.     

On Primes and Powers of a Fixed Integer

According to a 1904 conjecture of Dickson [2] unless preventedby congruence conditions, any finite collection of linear formsin Z[x] with positive leading coefficients infinitely oftensimultaneously represent primes. For the forms x, x+2 this includesthe conjectured infinitude of prime-pairs.     

On Fusion in Unipotent Blocks

The context of this note is as follows. One considers a connectedreductive group G and a Frobenius endomorphism F: G G definingG over a finite field of order q. One denotes by G^F the associated(finite) group of fixed points. Let l be a prime not dividing q. We are interested in the l-blocksof the finite group G^F. Such a block is called unipotent ifthere is a unipotent character (see, for instance, [6, Definition12.1]) among its representations in characteristic zero. Roughlyspeaking, it is believed that the study of arbitrary blocksof G^F might be reduced to unipotent blocks (see [2, Théorème2.3], [5, Remark 3.6]). In view of certain conjectures aboutblocks (see, for instance, [9]), it would be interesting tofurther reduce the study of unipotent blocks to the study ofprincipal blocks (blocks containing the trivial character).Our Theorem 7 is a step in that direction: we show that thelocal structure of any unipotent block of G^F is very close tothat of a principal block of a group of related type (notionof ‘control of fusion’, see [13, 49]). 1991 MathematicsSubject Classification 20Cxx.     

Simultaneous prime specializations of polynomials over finite fields

Recently the author proposed a uniform analogue of the Bateman–Hornconjectures for polynomials with coefficients from a finitefield (that is, for polynomials in F_q[T] rather than Z[T]).Here we use an explicit form of the Chebotarev density theoremover function fields to prove this conjecture in particularranges of the parameters. We give some applications includingthe solution of a problem posed by Hall.     

Locally Nilpotent p-Groups whose Proper Subgroups are Hypercentral or Nilpotent-by-Chernikov

Let G be a group and P be a property of groups. If every propersubgroup of G satisfies P but G itself does not satisfy it,then G is called a minimal non-P group. In this work we studylocally nilpotent minimal non-P groups, where P stands for ‘hypercentral’or ‘nilpotent-by-Chernikov’. In the first case weshow that if G is a minimal non-hypercentral Fitting group inwhich every proper subgroup is solvable, then G is solvable(see Theorem 1.1 below). This result generalizes [3, Theorem1]. In the second case we show that if every proper subgroupof G is nilpotent-by-Chernikov, then G is nilpotent-by-Chernikov(see Theorem 1.3 below). This settles a question which was consideredin [1–3, 10]. Recently in [9], the non-periodic case ofthe above question has been settled but the same work containsan assertion without proof about the periodic case. The main results of this paper are given below (see also [13]).     

Kazhdan-Lusztig Cells, q-Schur Algebras and James' Conjecture

We consider the Dipper–James q-Schur algebra S_q(n, r)_k,defined over a field k and with parameter q 0. An understandingof the representation theory of this algebra is of considerableinterest in the representation theory of finite groups of Lietype and quantum groups; see, for example, [6] and [11]. Itis known that S_q(n, r)_k is a semisimple algebra if q is nota root of unity. Much more interesting is the case when S_q(n,r)_k is not semisimple. Then we have a corresponding decompositionmatrix which records the multiplicities of the simple modulesin certain ‘standard modules’ (or ‘Weyl modules’).A major unsolved problem is the explicit determination of thesedecomposition matrices.     

Rates of Recurrence for Zq and Rq Extensions of Subshifts of Finite Type

In this note we give new asymptotic formulae for certain countingfunctions associated to the periodic behaviour of Z^q and R^qextensions of subshifts of finite type. In the case of the Z^qextensions, these strengthen previous estimates of Marcus andTuncel [9]. For both types of extension, our results complementthe central limit type results of Lalley [6]. Our proof requiresthe application of ideas from thermodynamic formalism. Whilstdeveloping this approach, in Section 2, we take the opportunityto present a counter-example to a related conjecture of Coelho-Filho[2].     

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.     

On the Analytic Order-Preserving Discrete-Time Dynamical Systems in Rn with every Fixed Point Stable

This paper studies the asymptotic behaviour of an analytic order-preservingdiscrete-time dynamical system in Rⁿ, which is usually generatedby a periodic cooperative system. The author proves that forsuch a dynamical system, if every fixed point is Liapunov stableand every positive semi-orbit has compact closure, then everypositive semi-orbit converges. This result does not requirethe assumption ‘strongly’ and gives an affirmativeanswer to the conjecture proposed by the author in [17] forthe analytic case.     

Examples of Non-Homeomorphic Harmonic Maps Between Negatively Curved Manifolds

Let M and N be closed non-positively curved manifolds, and letf:MN be a smooth map. Results of Eells and Sampson [1] showthat f is homotopic to a harmonic map , and Hartman [6] showedthat this is unique when N is negatively curved and f_*(₁ M)is not cyclic. Lawson and Yau conjectured that if f:MN is ahomotopy equivalence, where M and N are negatively curved, thenthe unique harmonic map homotopic to f would be a diffeomorphism.Counterexamples to this conjecture appeared in [2], and laterin [7] and [5]. There remains the question of whether a ‘topological’Lawson–Yau conjecture holds. 1991 Mathematics SubjectClassification 53C20, 55P10, 57C25, 58E20.     

Semisimple Modules for Finite Groups of Lie Type

Let k be an algebraically closed field of characteristic p >0, and let G be a connected, reductive algebraic group overk. In [8] and [11], conditions on the dimension of rationalG modules were seen to imply semisimplicity of these modules.In [8], certain of these conditions were extended to cover thefinite groups of Lie type. In this paper, we extend some ofthe results of [11] to cover these finite Lie type groups. Themain such extension is the following result.     

Capillary waves past a flat plate in water of finite depth

Free-surface flow past a semi-infinite flat plate in a channelof finite depth is considered. The fluid is assumed to be inviscidand incompressible, and the flow to be two-dimensional and irrotational.Surface tension is included in the dynamic boundary conditionbut the effects of gravity are neglected. It is shown that thereis a three-parameter family of solutions with waves in the farfield and a discontinuity in slope at the separation point.This family includes as particular cases the solutions previouslycomputed by Osborn & Stump (2001, Phys. Fluids, 13, 616–623)and by Andersson & Vanden-Broeck (1996, Proc. R. Soc., 452,1985–1997).     

Hammocks and the Nazarova-Roiter Algorithm

Hammocks have been considered by Brenner [1], who gave a numericalcriterion for a finite translation quiver to be the Auslander–Reitenquiver of some representation-finite algebra. Ringel and Vossieck[11] gave a combinatorial definition of left hammocks whichgeneralised the concept of hammocks in the sense of Brenner,as a translation quiver H and an additive function h on H (calledthe hammock function) satisfying some conditions. They showedthat a thin left hammock with finitely many projective verticesis just the preprojective component of the Auslander–Reitenquiver of the category of S-spaces, where S is a finite partiallyordered set (abbreviated as ‘poset’). An importantrole in the representation theory of posets is played by twodifferentiation algorithms. One of the algorithms was developedby Nazarova and Roiter [8], and it reduces a poset S with amaximal element a to a new poset S'=_aS. The second algorithmwas developed by Zavadskij [13], and it reduces a poset S witha suitable pair (a, b) of elements a, b to a new poset S'=_(a,b)S.The main purpose of this paper is to construct new left hammocksfrom a given one, and to show the relationship between thesenew left hammocks and the Nazarova–Roiter algorithm. Ina later paper [5], we discuss the relationship between hammocksand the Zavadskij algorithm.     

Varieties Which are almost D-Affine

This paper produces several examples of varieties X for whichthe global sections functor (X,–): D_X-modD(X)-mod is exact,and makes D(X)-mod a quotient category of D_X-mod, but is notan equivalence. These varieties are quotients by finite groupactions of D-affine varieties. The torsion of (X,–) isalso described, in some cases. Here, D_x-mod denotes the categoryof quasi-coherent D_X-modules.     

A Generalised Skolem-Mahler-lech Theorem for Affine Varieties

The Skolem–Mahler–Lech theorem states that if f(n)is a sequence given by a linear recurrence over a field of characteristic0, then the set of m such that f(m) is equal to 0 is the unionof a finite number of arithmetic progressions in m 0 and afinite set. We prove that if X is a subvariety of an affinevariety Y over a field of characteristic 0 and q is a pointin Y, and is an automorphism of Y, then the set of m such that^m(q) lies in X is a union of a finite number of complete doubly-infinitearithmetic progressions and a finite set. We show that thisis a generalisation of the Skolem–Mahler–Lech theorem.     

Beardon's Diophantine Equations and Non-Free Mobius Groups

Let µ be a real number. The Möbius group G_µis the matrix group generated by It is known that G_µ is free if |µ| 2 (see [1])or if µ is transcendental (see [3, 8]). Moreover, thereis a set of irrational algebraic numbers µ which is densein (–2, 2) and for which G_µ is non-free [2, p. 528].We may assume that µ > 0, and in this paper we considerrational µ in (0, 2). The following problem is difficult. Let G^nf denote the set of all rational numbers µ in (0,2) for which G_µ is non-free. In 1969 Lyndon and Ullman[8] proved that G^nf contains the elements of the forms p/(p²+ 1) and 1/(p + 1), where p = 1, 2, ..., and that if µ₀ G^nf, then µ₀/p G^nf for p = 1, 2, .... In 1993 Beardon[2] studied problem (P) by means of the words of the form A^rB^s A^t and A^r B^s A^t B^u A^v, and he obtained a sufficient conditionfor solvability of (P), included implicitly in [2, pp. 530–531],by means of the following Diophantine equations: 1991 Mathematics SubjectClassification 20E05, 20H20, 11D09.