Finite branch solutions to Painlevé VI around a fixed singular point

Every finite branch local solution to the sixth Painlevé equation around a fixed singular point is an algebraic branch solution. In particular a global solution is an algebraic solution if and only if it is finitely many-valued globally. The proof of this result relies on algebraic geometry of Painlevé VI, Riemann-Hilbert correspondence, geometry and dynamics on cubic surfaces, resolutions of Kleinian singularities, and power geometry of algebraic differential equations. In the course of the proof we are also able to classify all finite branch solutions up to Bäcklund transformations.     

Broue's Conjecture for the Hall-Janko Group and its Double Cover

Broué's abelian defect conjecture suggests a deep linkbetween the module categories of a block of a group algebraand its Brauer correspondent, viz. that they should be derivedequivalent. We are able to verify Broué's conjecturefor the Hall–Janko group, even its double cover 2.J₂,as well as for U₃(4) and Sp₄(4). In fact we verify Rickard'srefinement to Broué's conjecture and show that the derivedequivalence can be chosen to be a splendid equivalence for theseexamples. 2000 Mathematical Subject Classification: 20C20, 20C34.     

A Calculation of the Culler-shalen Seminorms Associated to Small Seifert Dehn Fillings

We determine the relationship between the multiplicities ofthe zeros of certain rational functions defined on the SL₂(C)and PSL₂C character varieties of knot exteriors. This leadsto a formula for the Culler–Shalen seminorms associatedto small Seifert Dehn fillings of such manifolds. 2000 MathematicsSubject Classification: 57M05, 57M27, 57R65.     

Kazhdan-Lusztig Cells and the Murphy Basis

Let H be the Iwahori–Hecke algebra associated with S_n,the symmetric group on n symbols. This algebra has two importantbases: the Kazhdan–Lusztig basis and the Murphy basis.We establish a precise connection between the two bases, allowingus to give, for the first time, purely algebraic proofs fora number of fundamental properties of the Kazhdan–Lusztigbasis and Lusztig's results on the a-function. 2000 MathematicsSubject Classification 20C08.     

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.     

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 .     

Relative K0, Annihilators, Fitting Ideals and Stickelberger Phenomena

When G is abelian and l is a prime we show how elements of therelative K-group K₀(Z_l[G], Q_l give rise to annihilator/Fittingideal relations of certain associated Z[G]-modules. Examplesof this phenomenon are ubiquitous. Particularly, we give examplesin which G is the Galois group of an extension of global fieldsand the resulting annihilator/Fitting ideal relation is closelyconnected to Stickelberger's Theorem and to the conjecturesof Coates and Sinnott, and Brumer. Higher Stickelberger idealsare defined in terms of special values of L-functions; whenthese vanish we show how to define fractional ideals, generalisingthe Stickelberger ideals, with similar annihilator properties.The fractional ideal is constructed from the Borel regulatorand the leading term in the Taylor series for the L-function.En route, our methods yield new proofs, in the case of abeliannumber fields, of formulae predicted by Lichtenbaum for theorders of K-groups and étale cohomology groups of ringsof algebraic integers. 2000 Mathematics Subject Classification11G55, 11R34, 11R42, 19F27.     

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.     

Monodromy of certain Painlevé–VI transcendents and reflection groups

We study the global analytic properties of the solutions of a particular family of Painlevé VI equations with the parameters β=γ=0, δ= and 2α=(2μ-1)² with arbitrary μ, 2μ≠∈ℤ. We introduce a class of solutions having critical behaviour of algebraic type, and completely compute the structure of the analytic continuation of these solutions in terms of an auxiliary reflection group in the three dimensional space. The analytic continuation is given in terms of an action of the braid group on the triples of generators of the reflection group. We show that the finite orbits of this action correspond to the algebraic solutions of our Painlevé VI equation and use this result to classify all of them. We prove that the algebraic solutions of our Painlevé VI equation are in one-to-one correspondence with the regular polyhedra or star-polyhedra in the three dimensional space. Oblatum 19-III-1999 & 25-XI-1999?Published online: 21 February 2000     

The streamline-diffusion method for nonconforming Qrot1 elements on rectangular tensor-product meshes

When the streamline–diffusion finite element method isapplied to convection–diffusion problems using nonconformingtrial spaces, it has previously been observed that stabilityand convergence problems may occur. It has consequently beenproposed that certain jump terms should be added to the bilinearform to obtain the same stability and convergence behaviouras in the conforming case. The analysis in this paper showsthat for the Q^rot₁ 1 element on rectangular shape-regular tensor-productmeshes, no jump terms are needed to stabilize the method. Inthis case moreover, for smooth solutions we derive in the streamline–diffusionnorm convergence of order h^3/2 (uniformly in the diffusion coefficientof the problem), where h is the mesh diameter. (This estimateis already known for the conforming case.) Our analysis alsoshows that similar stability and convergence results fail tohold true for analogous piecewise linear nonconforming elements.     

Local Cohomology of BP*BP-Comodules

Given a spectrum X, we construct a spectral sequence of BP_*BP-comodulesthat converges to BP_*(L_nX), where L_nX is the Bousfield localizationof X with respect to the Johnson–Wilson theory E(n)_*.The E₂-term of this spectral sequence consists of the derivedfunctors of an algebraic version of L_n. We show how to calculatethese derived functors, which are closely related to local cohomologyof BP_*-modules with respect to the ideal I_n+1. 2000 MathematicsSubject Classification 55N22, 55P60, 16W30.     

Totaro's Question on Zero-Cycles on G2, F4 and E6 Torsors

In a 2004 paper, Totaro asked whether a G-torsor X that hasa zero-cycle of degree d > 0 will necessarily have a closedétale point of degree dividing d, where G is a connectedalgebraic group. This question is closely related to severalconjectures regarding exceptional algebraic groups. Totaro gavea positive answer to his question in the following cases: Gsimple, split, and of type G₂, type F₄, or simply connectedof type E₆. We extend the list of cases where the answer is‘yes’ to all groups of type G₂ and some nonsplitgroups of type F₄ and E₆. No assumption on the characteristicof the base field is made. The key tool is a lemma regardinglinkage of Pfister forms.     

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 Number of Solutions of an Equation Over a Finite Field

The Stöhr–Voloch approach is used to obtain a newbound for the number of solutions in (F_q)² of an equation f(X,Y) = 0, where f(X, Y) is an absolutely irreducible polynomialwith coefficients in a finite field F_q.     

Hurwitz Groups with Given Centre

A Hurwitz group is any non-trivial finite group that can be(2,3,7)-generated; that is, generated by elements x and y satisfyingthe relations x² = y³ = (xy)⁷ = 1. In this short paper a completeanswer is given to a 1965 question by John Leech, showing thatthe centre of a Hurwitz group can be any given finite abeliangroup. The proof is based on a recent theorem of Lucchini, Tamburiniand Wilson, which states that the special linear group SL_n(q)is a Hurwitz group for every integer n 287 and every prime-powerq. 2000 Mathematics Subject Classification 20F05 (primary);57M05 (secondary).     

On regularity of a quasiconformal mapping at a point of maximal stretching

Local behaviour of a K-quasiconformal mapping f at a point z₀of maximal stretching is studied. A sufficient condition forthe existence of the finite limit lim(f(z) – f(z₀))/(z– z₀)|z – z₀|^1/K–1 as z z₀, and a criterionfor z₀ to be a point of maximal stretching are given.     

Mixed hp-DGFEM for incompressible flows II: Geometric edge meshes

We consider the Stokes problem of incompressible fluid flowin three-dimensional polyhedral domains discretized on hexahedralmeshes with hp-discontinuous Galerkin finite elements of typeQ_k for the velocity and Q_k–1 for the pressure. We provethat these elements are inf-sup stable on geometric edge meshesthat are refined anisotropically and non-quasiuniformly towardsedges and corners. The discrete inf-sup constant is shown tobe independent of the aspect ratio of the anisotropic elementsand is of O(k^–3/2) in the polynomial degree k, as in thecase of conforming Q_k–Q_k–2 approximations on thesame meshes.     

Value distribution of meromorphic solutions of certain difference Painlevé equations

The Borel exceptional value and the exponents of convergence of poles, zeros and fixed points of finite order transcendental meromorphic solutions for difference Painlevé I and II equations are estimated. And the forms of rational solutions of the difference Painlevé II equation and the autonomous difference Painlevé I equation are also given. It is also proved that the non-autonomous difference Painlevé I equation has no rational solution.     

The A-Polynomial has Ones in the Corners

1. Definition of the A-polynomial The A-polynomial was introduced in [3] (see also [5]), and wepresent an alternative definition here. Let M be a compact 3-manifoldwith boundary a torus T. Pick a basis , µ of ₁T, whichwe shall refer to as the longitude and meridian. Consider thesubset R_U of the affine algebraic variety R = Hom (₁M, SL₂C)having the property that () and (µ) are upper triangular.This is an algebraic subset of R, since one just adds equationsstating that the bottom-left entries in certain matrices arezero. There is a well-defined eigenvalue map given by taking the top-left entries of () and (µ).1991 Mathematics Subject Classification 57M25, 57M50.     

On Generators of Ideals Associated with Unions of Linear Varieties

Consider the polynomial ring R[x₁, ..., x_n] over a unique factorizationdomain R. A form (i.e., a homogeneous polynomial) is said tosplit if it is a product of linear forms. When a homogeneousideal is generated by splitting forms, the associated projectivealgebraic set is a finite union of linear subvarieties of P^n–1(R).But conversely, when a projective algebraic set decomposes intolinear subvarieties, its associated radical ideal may not begenerated by splitting forms. In this paper we construct a recursivealgorithm for establishing sufficient conditions for an idealto be generated by a prescribed set of splitting forms and applythis algorithm to a family of ideals that have arisen in thestudy of block designs. Our results on ideal generators havevery interesting applications to graph theory, which are discussedelsewhere.