Orthogonal, symplectic and unitary representations of finite groups
Carl
R.
Riehm
4687-4727
Abstract: Let $K$ be a field, $G$ a finite group, and $\rho: G \to \mathbf{GL}(V)$ a linear representation on the finite dimensional $K$-space $V$. The principal problems considered are: I. Determine (up to equivalence) the nonsingular symmetric, skew symmetric and Hermitian forms $h: V \times V \rightarrow K$ which are $G$-invariant. II. If $h$ is such a form, enumerate the equivalence classes of representations of $G$ into the corresponding group (orthogonal, symplectic or unitary group). III. Determine conditions on $G$ or $K$ under which two orthogonal, symplectic or unitary representations of $G$ are equivalent if and only if they are equivalent as linear representations and their underlying forms are ``isotypically'' equivalent. This last condition means that the restrictions of the forms to each pair of corresponding isotypic (homogeneous) $KG$-module components of their spaces are equivalent. We assume throughout that the characteristic of $K$ does not divide $2\vert G\vert$. Solutions to I and II are given when $K$ is a finite or local field, or when $K$ is a global field and the representation is ``split''. The results for III are strongest when the degrees of the absolutely irreducible representations of $G$ are odd - for example if $G$ has odd order or is an Abelian group, or more generally has a normal Abelian subgroup of odd index - and, in the case that $K$ is a local or global field, when the representations are split.
Representation type of $q$-Schur algebras
Karin
Erdmann;
Daniel
K.
Nakano
4729-4756
Abstract: In this paper we classify the $q$-Schur algebras having finite, tame or wild representation type and also the ones which are semisimple.
Geometry of chain complexes and outer automorphisms under derived equivalence
Birge
Huisgen-Zimmermann;
Manuel
Saorín
4757-4777
Abstract: The two main theorems proved here are as follows: If $A$ is a finite dimensional algebra over an algebraically closed field, the identity component of the algebraic group of outer automorphisms of $A$ is invariant under derived equivalence. This invariance is obtained as a consequence of the following generalization of a result of Voigt. Namely, given an appropriate geometrization $\operatorname{Comp}^{A}_{{\mathbf d}}$ of the family of finite $A$-module complexes with fixed sequence ${\mathbf{d}}$ of dimensions and an ``almost projective'' complex $X\in \operatorname{Comp}^{A} _{{\mathbf d}}$, there exists a canonical vector space embedding \begin{displaymath}T_{X}(\operatorname{Comp}^{A}_{{\mathbf{d}}}) / T_{X}(G.X) \... ...atorname{Hom} _{D^{b}(A{\operatorname{\text{-}Mod}})}(X,X[1]), \end{displaymath} where $G$ is the pertinent product of general linear groups acting on $\operatorname{Comp}^{A}_{{\mathbf{d}}}$, tangent spaces at $X$ are denoted by $T_{X}(-)$, and $X$ is identified with its image in the derived category $D^{b} (A{\operatorname{\text{-}Mod}})$.
Random variable dilation equation and multidimensional prescale functions
Julie
Belock;
Vladimir
Dobric
4779-4800
Abstract: A random variable $Z$ satisfying the random variable dilation equation $MZ \overset{d}{=}Z+G$, where $G$ is a discrete random variable independent of $Z$ with values in a lattice $\Gamma \subset$ $\mathbf{R}^{d}$ and weights $\left\{ c_{k}\right\} _{k\in \Gamma }$ and $M$ is an expanding and $\Gamma$-preserving matrix, if absolutely continuous with respect to Lebesgue measure, will have a density $\varphi$ which will satisfy a dilation equation \begin{displaymath}\varphi \left( x\right) =\left\vert \det M\right\vert \sum_{k\in \Gamma} c_{k}\varphi \left( Mx-k\right) \text{.} \end{displaymath} We have obtained necessary and sufficient conditions for the existence of the density $\varphi$ and a simple sufficient condition for $\varphi$'s existence in terms of the weights $\left\{ c_{k}\right\} _{k\in \Gamma }.$Wavelets in $\mathbf{R}^{d}$ can be generated in several ways. One is through a multiresolution analysis of $L^{2}\left( \mathbf{R}^{d}\right)$ generated by a compactly supported prescale function $\varphi$. The prescale function will satisfy a dilation equation and its lattice translates will form a Riesz basis for the closed linear span of the translates. The sufficient condition for the existence of $\varphi$ allows a tractable method for designing candidates for multidimensional prescale functions, which includes the case of multidimensional splines. We also show that this sufficient condition is necessary in the case when $\varphi$ is a prescale function.
Second class particles as microscopic characteristics in totally asymmetric nearest-neighbor $K$-exclusion processes
Timo
Seppäläinen
4801-4829
Abstract: We prove laws of large numbers for a second class particle in one-dimensional totally asymmetric $K$-exclusion processes, under hydrodynamic Euler scaling. The assumption required is that initially the ambient particle configuration converges to a limiting profile. The macroscopic trajectories of second class particles are characteristics and shocks of the conservation law of the particle density. The proof uses a variational representation of a second class particle, to overcome the problem of lack of information about invariant distributions. But we cannot rule out the possibility that the flux function of the conservation law may be neither differentiable nor strictly concave. To give a complete picture we discuss the construction, uniqueness, and other properties of the weak solution that the particle density obeys.
Centered complexity one Hamiltonian torus actions
Yael
Karshon;
Susan
Tolman
4831-4861
Abstract: We consider symplectic manifolds with Hamiltonian torus actions which are ``almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are ``centered" and the moment map is proper. In particular, this classifies the preimages under the moment map of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmannians $\operatorname{Gr}^+(2,\mathbb R^5)$ and $\operatorname{Gr}^+(2,\mathbb R^6)$ by two equal symplectic balls.
A measurable cardinal with a closed unbounded set of inaccessibles from $o(\kappa)=\kappa$
William
Mitchell
4863-4897
Abstract: We prove that $o(\kappa)=\kappa$ is sufficient to construct a model $V[C]$in which $\kappa$ is measurable and $C$ is a closed and unbounded subset of $\kappa$ containing only inaccessible cardinals of $V$. Gitik proved that $o(\kappa)=\kappa$ is necessary. We also calculate the consistency strength of the existence of such a set $C$ together with the assumption that $\kappa$ is Mahlo, weakly compact, or Ramsey. In addition we consider the possibility of having the set $C$ generate the closed unbounded ultrafilter of $V$ while $\kappa$ remains measurable, and show that Radin forcing, which requires a weak repeat point, cannot be improved on.
Automorphisms of the lattice of $\Pi_1^0$ classes; perfect thin classes and anc degrees
Peter
Cholak;
Richard
Coles;
Rod
Downey;
Eberhard
Herrmann
4899-4924
Abstract: $\Pi_1^0$ classes are important to the logical analysis of many parts of mathematics. The $\Pi_1^0$ classes form a lattice. As with the lattice of computably enumerable sets, it is natural to explore the relationship between this lattice and the Turing degrees. We focus on an analog of maximality, or more precisely, hyperhypersimplicity, namely the notion of a thin class. We prove a number of results relating automorphisms, invariance, and thin classes. Our main results are an analog of Martin's work on hyperhypersimple sets and high degrees, using thin classes and anc degrees, and an analog of Soare's work demonstrating that maximal sets form an orbit. In particular, we show that the collection of perfect thin classes (a notion which is definable in the lattice of $\Pi_1^0$ classes) forms an orbit in the lattice of $\Pi_1^0$classes; and a degree is anc iff it contains a perfect thin class. Hence the class of anc degrees is an invariant class for the lattice of $\Pi_1^0$classes. We remark that the automorphism result is proven via a $\Delta_3^0$automorphism, and demonstrate that this complexity is necessary.
Conditions imposed by tacnodes and cusps
Joaquim
Roé
4925-4948
Abstract: The study of linear systems of algebraic plane curves with fixed imposed singularities is a classical subject which has recently experienced important progress. The Horace method introduced by A. Hirschowitz has been successfully exploited to prove many $H^1$-vanishing theorems, even in higher dimension. Other specialization techniques, which include degenerations of the plane, are due to Z. Ran and C. Ciliberto and R. Miranda. G. M. Greuel, C. Lossen and E. Shustin use a local specialization procedure together with the Horace method to give the first asymptotically proper general existence criterion for singular curves of low degree. In this paper we develop a specialization method which allows us to compute the dimension of several linear systems as well as to substantially improve the bounds given by Greuel, Lossen and Shustin for curves with tacnodes and cusps.
The curve of ``Prym canonical'' Gauss divisors on a Prym theta divisor
Roy
Smith;
Robert
Varley
4949-4962
Abstract: The Gauss linear system on the theta divisor of the Jacobian of a nonhyperelliptic curve has two striking properties: 1) the branch divisor of the Gauss map on the theta divisor is dual to the canonical model of the curve; 2) those divisors in the Gauss system parametrized by the canonical curve are reducible. In contrast, Beauville and Debarre prove on a general Prym theta divisor of dimension $\ge 3$ all Gauss divisors are irreducible and normal. One is led to ask whether properties 1) and 2) may characterize the Gauss system of the theta divisor of a Jacobian. Since for a Prym theta divisor, the most distinguished curve in the Gauss system is the Prym canonical curve, the natural analog of the canonical curve for a Jacobian, in the present paper we analyze whether the analogs of properties 1) or 2) can ever hold for the Prym canonical curve. We note that both those properties would imply that the general Prym canonical Gauss divisor would be nonnormal. Then we find an explicit geometric model for the Prym canonical Gauss divisors and prove the following results using Beauville's singularities criterion for special subvarieties of Prym varieties: Theorem. For all smooth doubly covered nonhyperelliptic curves of genus $g\ge 5$, the general Prym canonical Gauss divisor is normal and irreducible. Corollary. For all smooth doubly covered nonhyperelliptic curves of genus $g\ge 4$, the Prym canonical curve is not dual to the branch divisor of the Gauss map.
On positivity of line bundles on Enriques surfaces
Tomasz
Szemberg
4963-4972
Abstract: We study linear systems on Enriques surfaces. We prove rationality of Seshadri constants of ample line bundles on Enriques surfaces and provide lower bounds on these numbers. We show the nonexistence of $k$-very ample line bundles on Enriques surfaces of degree $4k+4$ for $k\geq 1$, thus answering an old question of Ballico and Sommese.
Canonical splittings of groups and 3-manifolds
Peter
Scott;
Gadde
A.
Swarup
4973-5001
Abstract: We introduce the notion of a `canonical' splitting over $\mathbb{Z}$ or $\mathbb{Z}\times\mathbb{Z}$ for a finitely generated group $G$. We show that when $G$ happens to be the fundamental group of an orientable Haken manifold $M$ with incompressible boundary, then the decomposition of the group naturally obtained from canonical splittings is closely related to the one given by the standard JSJ-decomposition of $M$. This leads to a new proof of Johannson's Deformation Theorem.
Replacing model categories with simplicial ones
Daniel
Dugger
5003-5027
Abstract: In this paper we show that model categories of a very broad class can be replaced up to Quillen equivalence by simplicial model categories.
The hit problem for the Dickson algebra
Nguyen
H. V.
Hu'ng;
Tran
Ngoc
Nam
5029-5040
Abstract: Let the mod 2 Steenrod algebra, $\mathcal{A}$, and the general linear group, $GL(k,{\mathbb{F} }_2)$, act on $P_{k}:={\mathbb{F} }_2[x_{1},...,x_{k}]$ with $\vert x_{i}\vert=1$ in the usual manner. We prove the conjecture of the first-named author in Spherical classes and the algebraic transfer, (Trans. Amer. Math Soc. 349 (1997), 3893-3910) stating that every element of positive degree in the Dickson algebra $D_{k}:=(P_{k})^{GL(k, {\mathbb{F} }_2)}$ is $\mathcal{A}$-decomposable in $P_{k}$ for arbitrary $k>2$. This conjecture was shown to be equivalent to a weak algebraic version of the classical conjecture on spherical classes, which states that the only spherical classes in $Q_0S^0$ are the elements of Hopf invariant one and those of Kervaire invariant one.
Diophantine approximation, Bessel functions and radially symmetric periodic solutions of semilinear wave equations in a ball
J.
Berkovits;
J.
Mawhin
5041-5055
Abstract: The aim of this paper is to consider the radially-symmetric periodic-Dirichlet problem on $[0,T] \times B^n[a]$ for the equation \begin{displaymath}u_{tt} - \Delta u = f(t,\vert x\vert,u),\end{displaymath} where $\Delta$ is the classical Laplacian operator, and $B^n[a]$ denotes the open ball of center $0$ and radius $a$ in ${\mathbb R}^n.$ When $\alpha = a/T$ is a sufficiently large irrational with bounded partial quotients, we combine some number theory techniques with the asymptotic properties of the Bessel functions to show that $0$ is not an accumulation point of the spectrum of the linear part. This result is used to obtain existence conditions for the nonlinear problem.
Monotonicity of stable solutions in shadow systems
Wei-Ming
Ni;
Peter
Polácik;
Eiji
Yanagida
5057-5069
Abstract: A shadow system appears as a limit of a reaction-diffusion system in which some components have infinite diffusivity. We investigate the spatial structure of its stable solutions. It is known that, unlike scalar reaction-diffusion equations, some shadow systems may have stable nonconstant (monotone) solutions. On the other hand, it is also known that in autonomous shadow systems any nonconstant non-monotone stationary solution is necessarily unstable. In this paper, it is shown in a general setting that any stable bounded (not necessarily stationary) solution is asymptotically homogeneous or eventually monotone in $x$.
La transition vers l'instabilité pour les ondes de choc multi-dimensionnelles
Denis
Serre
5071-5093
Abstract: We consider multi-dimensional shock waves. We study their stability in Hadamard's sense, following Erpenbeck and Majda's strategy. When the unperturbed shock is close to a Lax shock which is already $1$-d unstable, we show, under a generic hypothesis, that it cannot be strongly stable. We also characterize strong instability in terms of a sign of an explicit quadratic form. In most cases, the instability under 1-d perturbations, which occurs for exceptional shock waves, characterizes a transition between weak stability and strong instability in the multi-dimensional setting. RÉSUMÉ. Nous considérons la stabilité des ondes de choc multi-dimensionnelles, en suivant la stratégie d'Erpenbeck et Majda. Lorsque le choc non perturbé est proche d'un choc de Lax longitudinalement instable, nous montrons, moyennant une hypothèse générique, que des ondes de surface sont présentes, empêchant ainsi la stabilité forte. Nous donnons aussi un critère d'instabilité forte en termes de signe d'une certaine forme quadratique. L'instabilité $1$-d d'un choc est en général facile à établir, car elle revêt un caractère exceptionnel. Elle apparaît comme une transition entre la stabilité faible et l'instabilité dans le contexte multi-d.
On the wellposedness of constitutive laws involving dissipation potentials
Wolfgang
Desch;
Ronald
Grimmer
5095-5120
Abstract: We consider a material with memory whose constitutive law is formulated in terms of internal state variables using convex potentials for the free energy and the dissipation. Given the stress at a material point depending on time, existence of a strain and a set of inner variables satisfying the constitutive law is proved. We require strong coercivity assumptions on the potentials, but none of the potentials need be quadratic. As a technical tool we generalize the notion of an Orlicz space to a cone ``normed'' by a convex functional which is not necessarily balanced. Duality and reflexivity in such cones are investigated.
Geometric representation of substitutions of Pisot type
Vincent
Canterini;
Anne
Siegel
5121-5144
Abstract: We prove that a substitutive dynamical system of Pisot type contains a factor which is isomorphic to a minimal rotation on a torus. If the substitution is unimodular and satisfies a certain combinatorial condition, we prove that the dynamical system is measurably conjugate to an exchange of domains in a self-similar compact subset of the Euclidean space.
Invariant measures for parabolic IFS with overlaps and random continued fractions
K.
Simon;
B.
Solomyak;
M.
Urbanski
5145-5164
Abstract: We study parabolic iterated function systems (IFS) with overlaps on the real line. An ergodic shift-invariant measure with positive entropy on the symbolic space induces an invariant measure on the limit set of the IFS. The Hausdorff dimension of this measure equals the ratio of entropy over Lyapunov exponent if the IFS has no ``overlaps.'' We focus on the overlapping case and consider parameterized families of IFS, satisfying a transversality condition. Our main result is that the invariant measure is absolutely continuous for a.e. parameter such that the entropy is greater than the Lyapunov exponent. If the entropy does not exceed the Lyapunov exponent, then their ratio gives the Hausdorff dimension of the invariant measure for a.e. parameter value, and moreover, the local dimension of the exceptional set of parameters can be estimated. These results are applied to a family of random continued fractions studied by R. Lyons. He proved singularity above a certain threshold; we show that this threshold is sharp and establish absolute continuity for a.e. parameter in some interval below the threshold.
Maximal semigroups in semi-simple Lie groups
Luiz
A. B.
San Martin
5165-5184
Abstract: The maximal semigroups with nonempty interior in a semi-simple Lie group with finite center are characterized as compression semigroups of subsets in the flag manifolds of the group. For this purpose a convexity theory, called here $\mathcal{B}$-convexity, based on the open Bruhat cells is developed. It turns out that a semigroup with nonempty interior is maximal if and only if it is the compression semigroup of the interior of a $\mathcal{B}$-convex set.