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.     

Hypersurface Singularities with 2-Dimensional Critical Locus

Consider an analytic germ f:(C^m, 0)(C, 0) (m3) whose criticallocus is a 2-dimensional complete intersection with an isolatedsingularity (icis). We prove that the homotopy type of the Milnorfiber of f is a bouquet of spheres, provided that the extendedcodimension of the germ f is finite. This result generalizesthe cases when the dimension of the critical locus is zero [8],respectively one [12]. Notice that if the critical locus isnot an icis, then the Milnor fiber, in general, is not homotopicallyequivalent to a wedge of spheres. For example, the Milnor fiberof the germ f:(C⁴, 0)(C, 0), defined by f(x₁, x₂, x₃, x₄) =x₁x₂x₃x₄ has the homotopy type of S¹xS¹xS¹. On the other hand,the finiteness of the extended codimension seems to be the rightgeneralization of the isolated singularity condition; see forexample [9–12, 17, 18]. In the last few years different types of ‘bouquet theorems’have appeared. Some of them deal with germs f:(X, x)(C, 0) wheref defines an isolated singularity. In some cases, similarlyto the Milnor case [8], F has the homotopy type of a bouquetof (dim X–1)-spheres, for example when X is an icis [2],or X is a complete intersection [5]. Moreover, in [13] Siersmaproved that F has a bouquet decomposition FF₀Sⁿ...Sⁿ (whereF₀ is the complex link of (X, x)), provided that both (X, x)and f have an isolated singularity. Actually, Siersma conjecturedand Tibr proved [16] a more general bouquet theorem for thecase when (X, x) is a stratified space and f defines an isolatedsingularity (in the sense of the stratified spaces). In thiscase F_iF_i, where the F_i are repeated suspensions of complexlinks of strata of X. (If (X, x) has the ‘Milnor property’,then the result has been proved by Lê; for details see[6].) In our situation, the space-germ (X, x) is smooth, but f hasbig singular locus. Surprisingly, for dim Sing f^–1(0)2,the Milnor fiber is again a bouquet (actually, a bouquet ofspheres, maybe of different dimensions). This result is in thespirit of Siersma's paper [12], where dim Sing f^–1(0)= 1. In that case, there is only a rather small topologicalobstruction for the Milnor fiber to be homotopically equivalentto a bouquet of spheres (as explained in Corollary 2.4). Inthe present paper, we attack the dim Sing f^–1(0) = 2 case.In our investigation some results of Zaharia are crucial [17,18].     

The Ubiquity of Thompson's Group F in Groups of Piecewise Linear Homeomorphisms of the Unit Interval

In the 1960s, Richard J. Thompson introduced a triple of groupsF T G which, among them, supplied the first examples of infinite,finitely presented, simple groups [14] (see [6] for publisheddetails), a technique for constructing an elementary exampleof a finitely presented group with an unsolvable word problem[12], the universal obstruction to a problem in homotopy theory[8], and the first examples of torsion free groups of type FPand not of type FP [5]. In abstract measure theory, it has beensuggested by Geoghegan (see [3] or [9, Question 13]) that Fmight be a counterexample to the conjecture that any finitelypresented group with no non-cyclic free subgroup is amenable(admits a bounded, non-trivial, finitely additive measure onall subsets that is invariant under left multiplication). Recently,F has arisen in the theory of groups of diagrams over semigrouppresentations [10], and as the object of questions in the algebraof string rewriting systems [7]. For more extensive bibliographiesand more results on Thompson's groups and their generalizationssee [1, 4, 6]. A persistent peculiarity of Thompson's groups is their abilityto pop up in diverse areas of mathematics. This suggests thatthere might be something very natural about Thompson's groups.We support this idea by showing (Theorem 1.1 below) that PL_o(I),the group of piecewise linear (finitely many changes of slope),orientation-preserving, self-homeomorphisms of the unit interval,is riddled with copies of F: a very weak criterion implies thata subgroup of PL_o(I) must contain an isomorphic copy of F.     

Discontinuous Homomorphisms from Non-Commutative Banach Algebras

In the 1970s, a question of Kaplansky about discontinuous homomorphismsfrom certain commutative Banach algebras was resolved. Let Abe the commutative C*-algebra C(), where is an infinite compactspace. Then, if the continuum hypothesis (CH) be assumed, thereis a discontinuous homomorphism from C() into a Banach algebra[2, 7]. In fact, let A be a commutative Banach algebra. Then(with (CH)) there is a discontinuous homomorphism from A intoa Banach algebra whenever the character space _A of A is infinite[3, Theorem 3] and also whenever there is a non-maximal, primeideal P in A such that |A/P|=2^₀ [4, 8]. (It is an open questionwhether or not every infinite-dimensional, commutative Banachalgebra A satisfies this latter condition.) 1991 MathematicsSubject Classification 46H40.     

Intertwining Maps from Certain Group Algebras

In [17, 18, 19], we began to investigate the continuity propertiesof homomorphisms from (non-abelian) group algebras. Alreadyin [19], we worked with general intertwining maps [3, 12]. Thesemaps not only provide a unified approach to both homomorphismsand derivations, but also have some significance in their ownright in connection with the cohomology comparison problem [4]. The present paper is a continuation of [17, 18, 19]; this timewe focus on groups which are connected or factorizable in thesense of [26]. In [26], G. A. Willis showed that if G is a connectedor factorizable, locally compact group, then every derivationfrom L¹(G) into a Banach L¹(G)-module is automatically continuous.For general intertwining maps from L¹(G), this conclusion isfalse: if G is connected and, for some nN, has an infinite numberof inequivalent, n-dimensional, irreducible unitary representations,then there is a discontinuous homomorphism from L¹(G into aBanach algebra by [18, Theorem 2.2] (provided that the continuumhypothesis is assumed). Hence, for an arbitrary intertwiningmap from L¹(G), the best we can reasonably hope for is a resultasserting the continuity of on a ‘large’, preferablydense subspace of L¹(G). Even if the target space of is a Banachmodule (which implies that the continuity ideal I() of is closed),it is not a priori evident that is automatically continuous:the proofs of the automatic continuity theorems in [26] relyon the fact that we can always confine ourselves to restrictionsto L¹(G) of derivations from M(G) [25, Lemmas 3.1 and 3.4].It is not clear if this strategy still works for an arbitraryintertwining map from L¹(G) into a Banach L¹(G)-module.     

Mixed finite element analysis of a non-linear three-fields Stokes model

In this article, a mixed finite element analysis of the non-linearStokes problem with monotone constitutive laws is considered.We construct a new three-field model for incompressible fluidswhere the velocity u, the non-linear stress tensor = (|u|)u and the pressure p are the most relevant unknowns. We giveexistence and unicity results for the continuous problem andits approximation. Stable and optimal error estimates underminimal regularity assumptions are derived and numerical resultsare presented. Received 29 April 1999. Accepted 30 November 1999.     

Weak Cayley Tables

In [1] Brauer puts forward a series of questions on group representationtheory in order to point out areas which were not well understood.One of these, which we denote by (B1), is the following: whatinformation in addition to the character table determines a(finite) group? In previous papers [5, 7–13], the originalwork of Frobenius on group characters has been re-examined andhas shed light on some of Brauer's questions, in particularan answer to (B1) has been given as follows. Frobenius defined for each character of a group G functions^(k):G^(k) C for k = 1, ..., deg with ⁽¹⁾ = . These functionsare called the k-characters (see [10] or [11] for their definition).The 1-, 2- and 3-characters of the irreducible representationsdetermine a group [7, 8] but the 1- and 2-characters do not[12]. Summaries of this work are given in [11] and [13].     

Quasilinear Elliptic Equations and Inequalities with Rapidly Growing Coefficients

Let us consider the boundary value problem where R^N is a bounded domain with smooth boundary (for example,such that certain Sobolev imbedding theorems hold). Let :RR, (s)=A(s²)s Then, if (s) = |s|^p–1s, p > 1, problem (1) is fairlywell understood and a great variety of existence results areavailable. These results are usually obtained using variationalmethods, monotone operator methods or fixed point and degreetheory arguments in the Sobolev space . If, on the other hand, we assume that is an oddnondecreasing function such that (0)=0, (t)>0, t>0, and is right continuous, then a Sobolev space setting for the problem is not appropriateand very general results are rather sparse. The first generalexistence results using the theory of monotone operators inOrlicz–Sobolev spaces were obtained in [5] and in [9,10]. Other recent work that puts the problem into this frameworkis contained in [2] and [8]. It is in the spirit of these latter papers that we pursue thestudy of problem (1) and we assume that F:xRR is a Carathéodoryfunction that satisfies certain growth conditions to be specifiedlater. We note here that the problems to be studied, when formulatedas operator equations, lead to the use of the topological degreefor multivalued maps (cf. [4, 16]). We shall see that a natural way of formulating the boundaryvalue problem will be a variational inequality formulation ofthe problem in a suitable Orlicz–Sobolev space. In orderto do this we shall have need of some facts about Orlicz–Sobolevspaces which we shall give now.     

A Necessary and Sufficient Condition for a Linear Differential System to be Strongly Monotone

In order to present the results of this note, we begin withsome definitions. Consider a differential system [formula] where IR is an open interval, and f(t, x), (t, x)IxRⁿ, is acontinuous vector function with continuous first derivativesf_r/x_s, r, s=1, 2, ..., n. Let D_xf(t, x), (t, x)IxRⁿ, denote the Jacobi matrix of f(t,x), with respect to the variables x₁, ..., x_n. Let x(t, t₀,x₀), tI(t₀, x₀) denote the maximal solution of the system (1)through the point (t₀, x₀)IxRⁿ. For two vectors x, yRⁿ, we use the notations x>y and x>>yaccording to the following definitions: [formula] An nxn matrix A=(a_rs) is called reducible if n2 and there existsa partition [formula] (p1, q1, p+q=n) such that [formula] The matrix A is called irreducible if n=1, or if n2 and A isnot reducible. The system (1) is called strongly monotone if for any t₀I, x₁,x₂Rⁿ [formula] holds for all t>t₀ as long as both solutions x(t, t₀, x_i),i=1, 2, are defined. The system is called cooperative if forall (t, x)IxRⁿ the off-diagonal elements of the nxn matrix D_xf(t,x) are nonnegative. 1991 Mathematics Subject Classification34A30, 34C99.     

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.     

On Power Series Having Sections with Multiply Positive Coefficients and a Theorem of Polya

Let (0.1) be a formal power series. In 1913, G. Pólya [7] provedthat if, for all sufficiently large n, the sections (0.2) have real negative zeros only, then the series (0.1) convergesin the whole complex plane C, and its sum f(z) is an entirefunction of order 0. Since then, formal power series with restrictionson zeros of their sections have been deeply investigated byseveral mathematicians. We cannot present an exhaustive bibliographyhere, and restrict ourselves to the references [1, 2, 3], wherethe reader can find detailed information. In this paper, we propose a different kind of generalisationof Pólya's theorem. It is based on the concept of multiplepositivity introduced by M. Fekete in 1912, and it has beentreated in detail by S. Karlin [4].     

Euler Characteristics of Real Varieties

In this paper we show how to associate to any real projectivealgebraic variety Z RP^n–1 a real polynomial F₁:Rⁿ,0 R, 0 with an algebraically isolated singularity, having theproperty that (Z) = (1 – deg (grad F₁), where deg (gradF₁ is the local real degree of the gradient grad F₁:Rⁿ, 0 Rⁿ,0. This degree can be computed algebraically by the method ofEisenbud and Levine, and Khimshiashvili [5]. The variety Z neednot be smooth. This leads to an expression for the Euler characteristic ofany compact algebraic subset of Rⁿ, and the link of a quasihomogeneousmapping f: Rⁿ, 0 Rⁿ, 0 again in terms of the local degree ofa gradient with algebraically isolated singularity. Similar expressions for the Euler characteristic of an arbitraryalgebraic subset of Rⁿ and the link of any polynomial map aregiven in terms of the degrees of algebraically finite gradientmaps. These maps do involve ‘sufficiently small’constants, but the degrees involved ar (theoretically, at least)algebraically computable.     

On Irregularities of Distribution II

Let a=(a₁, a₂, a₃, ...) be an arbitrary infinite sequence inU=[0, 1). Let Van der Corput [5] conjectured that d(a, n) (n=1, 2, ...) isunbounded, and this was proved in 1945 by van Aardenne-Ehrenfest[1]. Later she refined this [2], obtaining for infinitely many n. Here and later c₁, c₂, ... denote positiveabsolute constants. In 1954, Roth [8] showed that the quantity is closely related to the discrepancy of a suitable point setin U².     

Abel Equations: Composition Conjectures and the Model Problem

We consider the Abel equation where p and q are polynomials, and a is a fixed constant. Wedenote the solution of (1) by y(x, c), where y(a, c) = c. Standardexistence theorems ensure that y(x, c) is well-defined and analyticin both its arguments, for c sufficiently small. If y(b, c)= c, then we call y(x, c) a periodic solution. Likewise, ify(b, c) c for all c close to 0, then we say that the systemhas a centre between a and b. The numbers a and b are not important;by a simple transformation, we can always choose a = 0 and b= 1, and we shall usually do so from now on. Abel equations arise in several circumstances, but perhaps themain reason for their recent study is connected to the familyof systems where M and N are homogeneous polynomials of the same degreen. A transformation due to Cerkas allows us to bring these systemsto the form (1), where p and q are now trigonometric polynomials.It is not hard to show that, setting a = 0 and b = 2, the definitionsof periodic solution and centre for (1) coincide with theirusual definitions in the planar system (2). There are also transformationsto Abel-type equations for more general systems; see [8, 10]. This trigonometric Abel equation has been used in a large numberof works in order to estimate the number of limit cycles orobtain centre conditions, as well as in more general investigationsrelating the derivatives of the return map with iterated integrals.However, studying system (2) in whatever form is by no meanseasy, and a natural question is to ask whether we can stillcapture the essence of this problem if we take p and q to bepolynomials, in the hope that the calculations will become easier.The recent series of investigations by Briskin, Françoiseand Yomdin [3, 4, 5] seems to indicate that this could be thecase. Our interest here is to see what conditions the existence ofa centre in (1) imposes on the defining equations. For easeof reference, we shall always denote the antiderivative of thepolynomials p and q as P and Q; that is, 2000 Mathematics SubjectClassification 34C25, 34C99.     

On Simultaneous Diagonal Inequalities

Let F₁, ..., F_t be diagonal forms of degree k with real coefficientsin s variables, and let be a positive real number. The solubilityof the system of inequalities |F₁(x)|<,...,|F_t(x)|< in integers x₁, ..., x_s has been considered by a number of authorsover the last quarter-century, starting with the work of Cook[9] and Pitman [13] on the case t = 2. More recently, Brüdernand Cook [8] have shown that the above system is soluble providedthat s is sufficiently large in terms of k and t and that theforms F₁, ..., F_t satisfy certain additional conditions. Whathas not yet been considered is the possibility of allowing theforms F₁, ..., F_t to have different degrees. However, with therecent work of Wooley [18,20] on the corresponding problem forequations, the study of such systems has become a feasible prospect.In this paper we take a first step in that direction by studyingthe analogue of the system considered in [18] and [20]. Let₁, ..., _s and µ₁, ..., µ_s be real numbers such thatfor each i either _i or µ_i is nonzero. We define the forms and consider the solubility of the system of inequalities in rational integers x₁, ..., x_s. Although the methods developedby Wooley [19] hold some promise for studying more general systems,we do not pursue this in the present paper. We devote most ofour effort to proving the following theorem.     

A continuation method for a weakly elliptic two-parameter eigenvalue problem

We show that the continuation method can be used to solve aweakly elliptic two-parameter eigenvalue problem. We generalizethe continuation method for a nonsymmetric eigenvalue problemAx = x by T. Y. Li, Z. Zeng and L. Cong (1992 SIAM J. Numer.Anal. 29, 229–248) to two-parameter problems.     

Limits Along Parallel Lines and the Classical Fine Topology

The fine topology on Rⁿ (n2) is the coarsest topology for whichall superharmonic functions on Rⁿ are continuous. We refer toDoob [11, 1.XI] for its basic properties and its relationshipto the notion of thinness. This paper presents several theoremsrelating the fine topology to limits of functions along parallellines. (Results of this nature for the minimal fine topologyhave been given by Doob – see [10, Theorem 3.1] or [11,1.XII.23] – and the second author [15].) In particular,we will establish improvements and generalizations of resultsof Lusin and Privalov [18], Evans [12], Rudin [20], Bagemihland Seidel [6], Schneider [21], Berman [7], and Armitage andNelson [4], and will also solve a problem posed by the latterauthors. An early version of our first result is due to Evans [12, p.234], who proved that, if u is a superharmonic function on R³,then there is a set ER²x{0}, of two-dimensional measure 0, suchthat u(x, y,·) is continuous on R whenever (x, y, 0)E.We denote a typical point of Rⁿ by X=(X' x), where X'R^n–1and xR. Let :RⁿR^n–1x{0} denote the projection map givenby (X', x) = (X', 0). For any function f:Rⁿ[–, +] andpoint X we define the vertical and fine cluster sets of f atX respectively by C_V(f;X)={l[–, +]: there is a sequence (t_m) of numbersin R\{x} such that t_mx and f(X', t_m)l}| and C_F(f;X)={l[–, +]: for each neighbourhood N of l in [–,+], the set f^–1(N) is non-thin at X}. Sets which are open in the fine topology will be called finelyopen, and functions which are continuous with respect to thefine topology will be called finely continuous. Corollary 1(ii)below is an improvement of Evans' result.     

Permutation Groups in o-Minimal Structures

In this paper we develop a structure theory for transitive permutationgroups definable in o-minimal structures. We fix an o-minimalstructure M, a group G definable in M, and a set and a faithfultransitive action of G on definable in M, and talk of the permutationgroup (G, ). Often, we are concerned with definably primitivepermutation groups (G, ); this means that there is no propernon-trivial definable G-invariant equivalence relation on ,so definable primitivity is equivalent to a point stabiliserG being a maximal definable subgroup of G. Of course, sinceany group definable in an o-minimal structure has the descendingchain condition on definable subgroups [23] we expect many questionson definable transitive permutation groups to reduce to questionson definably primitive ones. Recall that a group G definable in an o-minimal structure issaid to be connected if there is no proper definable subgroupof finite index. In some places, if G is a group definable inM we must distinguish between definability in the full ambientstructure M and G-definability, which means definability inthe pure group G:= (G, .); for example, G is G-definably connectedmeans that G does not contain proper subgroups of finite indexwhich are definable in the group structure. By definable, wealways mean definability in M. In some situations, when thereis a field R definable in M, we say a set is R-semialgebraic,meaning that it is definable in (R, +, .). We call a permutationgroup (G, ) R-semialgebraic if G, and the action of G on canall be defined in the pure field structure of a real closedfield R. If R is clear from the context, we also just write‘semialgebraic’.     

The Pontryagin Rings of Moduli Spaces of Arbitrary Rank Holomorphic Bundles Over a Riemann Surface

The cohomology of M(n, d), the moduli space of stable holomorphicbundles of coprime rank n and degree d and fixed determinant,over a Riemann surface of genus g 2, has been widely studiedfrom a wide range of approaches. Narasimhan and Seshadri [17]originally showed that the topology of M(n, d) depends onlyon the genus g rather than the complex structure of . An inductivemethod to determine the Betti numbers of M(n, d) was first givenby Harder and Narasimhan [7] and subsequently by Atiyah andBott [1]. The integral cohomology of M(n, d) is known to haveno torsion [1] and a set of generators was found by Newstead[19] for n = 2, and by Atiyah and Bott [1] for arbitrary n.Much progress has been made recently in determining the relationsthat hold amongst these generators, particularly in the ranktwo, odd degree case which is now largely understood. A setof relations due to Mumford in the rational cohomology ringof M(2, 1) is now known to be complete [14]; recently severalauthors have found a minimal complete set of relations for the‘invariant’ subring of the rational cohomology ofM(2, 1) [2, 13, 20, 25]. Unless otherwise stated all cohomology in this paper will haverational coefficients.     

On Prime Ends and Plane Continua

Let f be a conformal map of the unit disk D onto the domainG = C{}. We shall always use the spherical metric in . Carathéodory [3] introduced the concept of a prime endof G in order to describe the boundary behaviour of f in geometricterms; see for example [6, Chapter 9] or [12, Section 2.4].There is a bijective map of T = D onto the set of prime ends of G.