We generalize the idea of reflexibility of a map on a surface by introducing certain integers as its ”exponents“. An exponent is any integer $e$ with the property that changing the cyclic permutation of edges around each vertex induced by the map to its $e$-th power gives rise to an isomorphic map. The exponents reduced modulo the least common multiple of the vertex valencies form an Abelian group, the exponent group $\mbox{Ex}\,(M)$ of the map $M$. Along with the automorphism group, the group in fact provides an additional measure of symmetry of ~$M$.

The paper is devoted to developing the fundamentals of the theory of exponent groups of maps. Motivation comes from the problem of classification of regular maps with a given underlying graph. To this end, we prove that the number of non-isomorphic regular maps (if any) with a given underlying graph and the same map automorphism group is $|{\Bbb Z}_n^*:\mbox{Ex}\,(M)|$, $n$ being the valency of $M$. We calculate the exponent groups for some interesting families of regular maps including complete maps. Special attention is paid to the problem of how the antipodality of a map is reflected by its exponent group. In the final section we discuss several open problems.

1991 Mathematics Subject Classification: 05C10, 05C25, 20F32.

Let $a_n$ be an increasing sequence of positive reals with $a_n \rightarrow \infty$ as $n \rightarrow \infty$. Necessary and sufficient conditions are obtained for each of the sequences $[\alpha a_n], [\alpha^{a_n}], [{a_n}^{\alpha}]$ to take on infinitely many prime values for almost all $\alpha > \beta$. For example, the sequence $[\alpha a_n]$ is infinitely often prime for almost all $\alpha > 0$ if and only if there is a subsequence of the $a_n$, say $b_n$, with $b_{n+1} > b_n + 1$ and with the series $\sum 1/{b_n}$ divergent. Asymptotic formulae are obtained when the sequences considered are lacunary. An earlier result of the author reduces the problem to estimating the measure of overlaps of certain sets, and sieve methods are used to obtain the correct order upper bounds.

1991 Mathematics Subject Classification: primary 11N05; secondary 11K99, 11N36.

The local structure of homomorphisms of commutative noetherian rings is investigated from the point of view of dualizing complexes. A concept of finite Gorenstein dimension, which substantially weakens the notion of finite flat dimension, is introduced for homomorphisms. It is shown to impose structural constraints, due to a remarkable equivalence of subcategories of the derived category of all modules.

An essential part of this study is the development of relative notions of dualizing complexes and Bass numbers. It is proved that the Bass numbers of local homomorphisms are rigid, extending a known result for local rings. Quasi-Gorenstein homomorphisms are introduced as local homomorphisms that base-change a dualizing complex for the source ring into one for the target. They are shown to have the stability properties of the Gorenstein homomorphism that they generalize.

1991 Mathematics Subject Classification: primary 13H10, 13D23, 14E40; secondary 13C15.

In this paper we find the multiplicities $\dim L(\lambda)_{\lambda-\alpha}$ where $\alpha$ is an {\em arbitrary} root and $L(\lambda)$ is an irreducible $SL_n$-module with highest weight $\lambda$. We provide different bases of the corresponding weight spaces and outline some applications to the symmetric groups. In particular we describe certain composition multiplicities in the modular branching rule.

1991 Mathematics Subject Classification: 20C05, 20G05.

In this paper a practical algorithm is given to find all binary forms with rational coefficients of given degree with discriminant divisible by a given finite set of rational primes, up to an obvious equivalence relation.

This is done by adapting the finiteness result of Evertse and Gy\H{o}ry. A technical assumption that all fields used have class number $1$ is made to aid the exposition.

All binary forms of degree less than or equal to $6$ with $2$-power discriminant are then calculated. This is then used to complete the table of curves of genus $2$ which had previously been computed by Merriman and the author. Some ranks of the associated Jacobian varieties are also computed.

1991 Mathematics Subject Classification: primary 11E76, 11G30; secondary 11Y40, 11D61.

Let $W$ be a finite dimensional representation of a linearly reductive group $G$ over a field $k$. Motivated by their work on classical rings of invariants, Levasseur and Stafford asked whether the ring of invariants under $G$ of the symmetric algebra of $W$ has a simple ring of differential operators.

In this paper, we show that this is true in prime characteristic. Indeed, if $R$ is a graded subring of a polynomial ring over a perfect field of characteristic $p>0$ and if the inclusion $R\hookrightarrow S$ splits, then $D_k(R)$ is a simple ring. In the last section of the paper, we discuss how one might try to deduce the characteristic zero case from this result. As yet, however, this is a subtle problem and the answer to the question of Levasseur and Stafford remains open in characteristic zero.

http://www.luc.ac.be/Research/Algebra

1991 Mathematics Subject Classification: 16S32, 16G60, 13A35.

Consider the group scheme

$$ R \mapsto G(R) := R \rtimes R^* = \left( \begin{array}{cc} 1 & R \\ 0 & R^* \end{array} \right) = \left( \begin{array}{cc} 1 & *\\ 0 & * \end{array} \right) \cap \mathrm{GL}_2(R) $$

where $R$ is an arbitrary commutative ring with $1 \neq 0$ and a unit $x \in R^*$ acts on $R$ by multiplication.

We will study the finiteness properties of subgroups of $G(\Oka_S)$ where $\Oka_S$ is an $S$-arithmetic subring of a global function field. The subgroups we are interested in are of the form $\Oka_S \rtimes Q$ where $Q$ is a subgroup of $\Oka_S^*$. The finiteness properties of these metabelian groups can be expressed in terms of the $\Sigma$-invariant due to R. Bieri and R. Strebel.

{\sc Theorem A.} {\it Let $S$ be a finite set of places of a global function field (regarded as normalized discrete valuations) and $\Oka_S$ the corresponding $S$-arithmetic ring. Let $Q$ be a subgroup of $\Oka_S^*$. Then $Q$ is finitely generated and for all integers $n \geq 1$ the following are equivalent:

\begin{enumerate} \item[\rm (1)] $\Oka_S \rtimes Q$ is of type FP$_n$;

\item[\rm (2)] $\Oka_S$ is $n$-tame as a $\ZZZ Q$-module;

\item[\rm (3)] each $p \in S$ restricts to a non-trivial homomorphism $p|_{Q} : Q \rightarrow \RRR$ and the set $\{ [p|_Q] \mid p \in S \} \subseteq \SSS(Q)$ is $n$-tame. \end{enumerate}

If these conditions hold for at least one $n \geq 1$ then the identity

$$ \Sigma_{\Oka_S}^c(Q,\ZZZ) = \{ [p|_Q] \mid p \in S \} $$

holds.} {\sc Theorem B.} {\it Let $r$ denote the rank of $Q$. Then the following hold: \begin{enumerate} \item[\rm (1)] the group $\Oka_S \rtimes Q$ is not of type FP$_{r+1}$;

\item[\rm (2)] if $Q$ has maximum rank $r=|S|-1$ then the group $\Oka_S \rtimes Q$ is of type FP$_r$.

\end{enumerate} In particular, $G(\Oka_S) = \Oka_S \rtimes \Oka_S^*$ is of type FP$_{|S|-1}$ but not of type FP$_{|S|}$.}

1991 Mathematics Subject Classification: 20E08, 20F16, 20G30, 52A20.

In this paper, we apply Padé approximation methods to derive completely explicit measures of irrationality for certain classes of algebraic numbers. Our approach is similar to that taken previously by G.V. Chudnovsky but has some fundamental advantages with regards to determining implicit constants. Our general results may be applied to produce specific bounds of the flavour of

$$ \left| \sqrt[3]{2} - \frac{p}{q} \right| > \frac{1}{4}~ q^{-2.45} \hskip2ex \mbox{and} \hskip2ex \left| \sqrt[7]{5} - \frac{p}{q} \right| > \frac{1}{4}~ q^{-4.43} $$

which we show to hold for any nonzero integers $p$ and $q$. Further examples are tabulated and applications to Diophantine equations are briefly discussed as are other topics of related interest.

1991 Mathematics Subject Classification: primary 11J68, 11J82; secondary 11D41.

Representations of Hecke and $q$-Schur algebras are closely related to those of finite general linear groups $G$ in non-describing characteristics. Such a relationship can be described by certain functors. Using these functors, we determine the Harish-Chandra vertices and sources of certain indecomposable $G$-modules. The Green correspondence is investigated in this context. As a further application of our theory, we establish Steinberg's tensor product theorems for irreducible representations of $G$ in non-describing characteristics.

1991 Mathematics Subject Classification: 20C20, 20C33, 20G05, 20G40.

In 1959 E. Bishop used a Banach-space version of the analytic duality principle established by e Silva, K\"{o}the, Grothendieck and others to study connections between spectral decomposition properties of a Banach-space operator and its adjoint. According to Bishop a continuous linear operator $T \in L(X)$ on a Banach space $X$ satisfies property $(\beta)$ if the multiplication operator ${\cal O}(U,X) \rightarrow {\cal O}(U,X),$ $f \mapsto (z-T)f,$ is injective with closed range for each open set $U$ in the complex plane. In the present article the analytic duality principle in its original locally convex form is used to develop a complete duality theory for property $(\beta)$. At the same time it is shown that, up to similarity, property $(\beta)$ characterizes those operators occurring as restrictions of operators decomposable in the sense of C. Foias, and that its dual property, formulated as a spectral decomposition property for the spectral subspaces of the given operator, characterizes those operators occurring as quotients of decomposable operators. It is proved that, unlike the situation for commuting subnormal operators, each finite commuting system of operators with property $(\beta)$ can be extended to a finite commuting system of decomposable operators. Meanwhile the results of this paper have been used to prove the existence of invariant subspaces for subdecomposable operators with sufficiently rich spectrum.

1991 Mathematics Subject Classification: 47A11, 47B40.

Let $\cal D $ denote the collection of dyadic intervals in the unit interval. Let $\tau$ be a rearrangement of the dyadic intervals. We study the induced operator

$$ Th_I = h_{\tau(I)}$$

where $h_I$ is the $L^{\infty}$ normalized Haar function. We find geometric conditions on $\tau$ which are necessary and sufficient for $T$ to be bounded on $BMO$. We also characterize the rearrangements of the $L^p$ normalized Haar system in $L^p.$

1991 Mathematics Subject Classification: 42C20, 43A17, 47B38.

A scheme of construction of infinite groups, other than simple groups, free groups of infinite rank and the infinite cyclic group, which are isomorphic to all their non-trivial normal subgroups is presented. Some results about the automorphism groups of simple infinite groups are also obtained. In particular, it is proved that there is an infinite group $G$ of any sufficiently large prime exponent $p$ (or which is torsion-free) all of whose proper subgroups are cyclic, and such that the groups ${\rm Aut}\,G$ and ${\rm Out}\,G$ are isomorphic. The proofs use the technique of graded diagrams developed by A. Yu. Ol'shanskii.

1991 Mathematics Subject Classification: 20F05, 20F06.

The purpose of this paper is to describe a general procedure for computing analogues of Young's seminormal representations of the symmetric groups. The method is to generalize the Jucys-Murphy elements in the group algebras of the symmetric groups to arbitrary Weyl groups and Iwahori-Hecke algebras. The combinatorics of these elements allows one to compute irreducible representations explicitly and often very easily. In this paper we do these computations for Weyl groups and Iwahori-Hecke algebras of types $A_n$, $B_n$, $D_n$, $G_2$. Although these computations are in reach for types $F_4$, $E_6$ and $E_7$, we shall postpone this to another work.

1991 Mathematics Subject Classification: primary 20F55, 20C15; secondary 20C30, 20G05.

We consider covering properties of weak topologies of Banach spaces, especially of weak or point-wise topologies of function spaces $C(K)$, for compact spaces $K$. We answer questions posed by A. V. Arkhangel'skii, S. P. Gul'ko, and R. W. Hansell. Our main results are the following. A Banach space of density at most $\omega_1$ is hereditarily metaLindel\" of in its weak topology. If the weight of a compact space $K$ is at most $\omega_1$, then the spaces $C_w(K)$ and $C_p(K)$ are hereditarily metaLindel\" of. Let $\bar T$ be the one-point compactification of a tree $T$. Then the space $C_p(\bar T)$ is hereditarily $\sigma$-metacompact. If $T$ is an infinitely branching full tree of uncountable height and of cardinality bigger than ${\bf c}$, then the weak topology of the unit sphere of $C(\bar T)$ is not $\sigma$-fragmented by any metric. The space $C_p(\beta\omega_1)$ is neither metaLindel\" of nor $\sigma$-relatively metacompact. The space $C_p(\beta\omega_2)$ is not $\sigma$-relatively metaLindel\" of. Under the set-theoretic axiom $\diamondsuit$, there exists a scattered compact space $K_1$ such that the space $C_p(K_1)$ is not $\sigma$-relatively metacompact, and under a related axiom \boxd, there exists a scattered compact space $K_2$ such that the space $C_p(K_2)$ is not $\sigma$-relatively metaLindel\" of.

1991 Mathematics Subject Classification: 54C35, 46B20, 54E20, 54D30.

In this paper we characterize those Banach spaces $E$ that admit a locally uniformly rotund renorming by means of a linear topological condition that is a particular case of the notion, introduced by J. E. Jayne, I. Namioka and C. A. Rogers, called countable cover by sets of small local diameter. This allows us to develop a Decomposition Method and a Transfer Technique to obtain locally uniformly rotund renormings.

1991 Mathematics Subject Classification: 46B20.

As shown by the author in {\em Proc.\ Amer.\ Math.\ Soc.} 115 (1992) 345–352, for every metric space $(K,d)$ with compact closed balls one has $(\mbox{lip}\,\varphi(K))^{**} = \mbox{Lip} \,\varphi (K),$ where $\varphi$ is any majorant (that is, non-decreasing function on ${\Bbb R}_+$ with $\varphi (0+) = \varphi (0) = 0$) such that $\varphi(t)/t$ monotonically tends to $+\infty$ as $t\rightarrow 0.$ Here $\mbox{Lip}\,\varphi(K)$ is the Lipschitz space on $K$ with respect to the metric $\varphi(d),\ \mbox{lip}\,\varphi(K)$ is the corresponding ’little‘ Lipschitz space of functions vanishing ’at infinity‘, and ’=‘ means ’canonically isometrically isomorphic‘. The main idea of the proof consisted of finding a normed space $M$ such that $M^* = \mbox{Lip}\,\varphi(K)$ and $M^c = (\mbox{lip}\,\varphi(K))^*,$ where ’$c$‘ stands for the completion, and identifying $M$ with the space of Borel measures on $K$ equipped with the Kantorovich norm. In the present paper, this argument is carried over to generalized Lipschitz spaces on ${\Bbb R}^n$ defined in terms of higher order differences. For an integer $k$ and for a majorant $\varphi$ with $\lim_{t \rightarrow 0}\varphi(t)/t ^k = +\infty,$ define $\Lambda ^k_\varphi$ to be the space of all bounded functions $f$ on ${\Bbb R}^n$ such that for some constant $C,\ \omega_k(f\,;t) \leq C\varphi(t)$ for all $t\geq 0$, where $\omega_k(f\,;\cdot)$ is the $k$th modulus of continuity of $f$. Let $\lambda ^k_\varphi$ be the (closed linear separable) subspace in $\Lambda ^k_\varphi$ which consists of functions $f$ vanishing at ’infinity' and such that $\lim_{t \rightarrow 0}\omega _k(f\,; t)/\varphi (t) = 0.$ We introduce an appropriate analogue $\Vert \cdot \Vert_{k,\varphi}$ of the Kantorovich norm on the space $M$ of finite Borel measures on ${\Bbb R}^n$ with compact support. The properties of this norm for $k \geq 2$ are significantly different from those of the Kantorovich norm ($k = 1$) which reflects the difference in analytic nature of generalized Lipschitz spaces versus classical ones. However, the core of the duality theory survives, and it is shown that $(M, \Vert\cdot \Vert _{k, \varphi})^* = \Lambda ^k _\varphi,\ \ (M, \Vert \cdot \Vert _{k,\varphi})^c = (\lambda ^k _\varphi)^*$ and consequently, $(\lambda ^k _\varphi )^{**} = \Lambda ^k _\varphi$. Several applications of these results are discussed, and a few open problems are formulated.

1991 Mathematics Subject Classification: 46E15, 46E35.

We construct a sequence of concordance invariants for classical links, which depend on the peripheral isomorphism type of the nilpotent quotients of the link fundamental group. The terminology stems from the fact that we replace the Magnus expansion in the definition of Milnor's μ-invariants by the similar Campbell-Hausdorff expansion. The main point is that we introduce a new universal indeterminacy, which depends only on the number of components of the link. The Campbell-Hausdorff invariants are new, effectively computable and can efficiently distinguish (unordered and unoriented) isotypes of links, as we indicate on several families of closed braid examples. They also satisfy certain natural dependence relations, which generalize well-known symmetries of the μ-invariants.

Let $f:({\mathbb R}^n,0)\rightarrow(\Bbb R^p,0)$ be a $C^\infty$ map-germ. We define $f$ to be finitely, or $\infty$-, $\mathcal A$-determined, if there exists an integer $m$ such that all germs $g$ with $j^mg(0)=j^mf(0)$, or if all germs $g$ with the same infinite Taylor series as $f$, respectively, are $\mathcal A$-equivalent to $f$. For any integer $k$, $0\le k<\infty$, we can consider $\mathcal A$\rq s $C^k$ counterpart (consisting of $C^k$ diffeomorphisms) $\mathcal A^{(k)}$, and we can define the notion of finite, or $\infty$-,$\mathcal A^{(k)}$-determinacy in a similar manner. Consider the following conditions for a $C^{\infty}$ germ $f$: (a$_k$) $f$ is $\infty$-$\mathcal A^{(k)}$-determined, (b$_k$) $f$ is finitely $\mathcal A^{(k)}$-determined, (t) $m_n^{\infty}\theta(f)\subset tf(m_n\theta(n))+\omega f(m_p\theta(p))$, (g) there exists a representative $ f:U\rightarrow\Bbb R^p$ defined on some neighbourhood $U$ of 0 in $\Bbb R^n$ such that the multigerm of $f$ is stable at every finite set $S\subset U-\lbrace0\rbrace$, and (g$^\prime$) every $f^\prime $ with $j^\infty f^\prime(0)=j^\infty f(0)$ satisfies condition (g). We also define a technical condition which will imply condition $(\mbox{g})$ above. This condition is a collection of $p+1$ \Lojasiewicz inequalities which express that the multigerm of $f$ is stable at any finite set of points outside 0 and only becomes unstable at a finite rate when we approach 0. We will denote this condition by (e). With this notation we prove the following. For any $ C^\infty$ map germ $f:(\Bbb R^n,0)\rightarrow(\Bbb R^p,0)$ the conditions (e), (t), (g$'$) and ($\mbox{a}_{\infty}$) are equivalent conditions. Moreover, each of these conditions is equivalent to any of ($\mbox{a}_k$) $(p+1\le k<\infty)$, ($\mbox{b}_k$) $(p+1\le k<\infty)$.

1991 Mathematics Subject Classification: 58C27.

Let $T$ be a contraction on a complex Hilbert space $H$. A result of Brown, Chevreau and Pearcy from 1979 shows that $T$ has a non-trivial invariant subspace if the spectrum of $T$ is dominating in the open unit disc. It is the purpose of the present paper to prove the multidimensional analogue of this result for spherical contractions $T \in L(H)^n$ that possess a spherical dilation and for which the Harte spectrum is dominating in the open unit ball $B$ in $\mathbb{C}^n$. If even the essential Harte spectrum of $T$ is dominating in $B$, then $T$ is shown to be reflexive and to possess an extremely rich invariant subspace lattice. The proof is based on the existence of an $H^{\infty}$-functional calculus for completely non-unitary spherical contractions and on a multidimensional analogue of the classical result of Sz.Nagy and Foias, stating that each spherical contraction which is neither of type $C_{\cdot 0}$ nor of type $C_{0 \cdot}$ and which does not consist of multiples of the identity operator on $H$, possesses non-trivial joint hyperinvariant subspaces.

1991 Mathematics Subject Classification: 47A13, 47A15, 47A60

When $m$ is odd, spreads in an orthogonal vector space of type $\Omega^+ (2m+2,2)$ are related to binary Kerdock codes and extremal line-sets in $\RR^{2^{m+1}}$ with prescribed angles. Spreads in a $2m$-dimensional binary symplectic vector space are related to Kerdock codes over $\ZZ_4$ and extremal line-sets in $\CC^{2^m}$ with prescribed angles. These connections involve binary, real and complex geometry associated with extraspecial 2-groups. A geometric map from symplectic to orthogonal spreads is shown to induce the Gray map from a corresponding $\ZZ_4$-Kerdock code to its binary image. These geometric considerations lead to the construction, for any odd composite $m$, of large numbers of $\ZZ_4$-Kerdock codes. They also produce new $\ZZ_4$-linear Kerdock and Preparata codes.

1991 Mathematics Subject Classification: primary 94B60; secondary 51M15, 20C99.