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 $W$ 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.

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.

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.

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.

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

We investigate a class of triangular UHF algebras which have the special property that there exists a sequence of unital multiplicative contractive finite-rank conditional expectations of the algebra into itself, which converges strongly to the identity, whose ranges form an increasing chain with dense union. A partial classification is given in terms of a local finiteness property, and a delimiting counterexample is obtained. Associated inverse limit algebras are considered, and a characterization of residually finite dimensional triangular AF algebras is obtained.

1991 Mathematics Subject Classification: 47D25, 46J35.

The Novikov-Landweber algebra and the Steenrod algebra are set up in terms of the primitive differential operators $D_k=\sum_i x_i^{k+1}\frac{\partial}{\partial x_i}$ acting in the usual way on the integral polynomial ring ${\bf Z}[x_1,\ldots ,x_n,\ldots]$. A commutative wedge product $\vee$ for differential operators is introduced and it is shown that the iterated wedge product $D_k^{\vee r}$ is divisible by $r!$ as an integral operator. The divided differential operator algebra $\cal D$ is generated over the integers by the divided operators $\frac{D_k^{\vee r}}{r!}$ under the wedge product. $\cal D$ is additively isomorphic to the abelian group of symmetric functions in the variables $x_i$. Furthermore $\cal D$ is closed under composition of operators and admits a natural coproduct which makes it a Hopf algebra in two ways, with respect to the composition and wedge products. Under composition $\cal D$ is isomorphic to the Landweber-Novikov algebra. A Hopf sub-algebra is generated under composition by the integral Steenrod squares $SQ^r = \frac{D_1^{\vee r}}{r!}$ and reduces mod 2 to the Steenrod algebra. An explicit product formula for two wedge expressions is developed and used to derive Milnor's product formula for his basis elements in the Steenrod algebra. The hit problem in the Steenrod algebra is reformulated in terms of partial differential operators.

1991 Mathematics Subject Classification 55S10.

It is known that a Markov map $T$ of the unit interval preserves a measure $\mu$, say, equivalent to Lebesgue measure, and that almost every point of the interval has a forward orbit under $T$ that is uniformly distributed with respect to $\mu$. In the opposite direction the main result of this paper states that there is a set of points having Hausdorff dimension $1$ whose forward orbits are in a certain sense very far from being so distributed.

1991 Mathematics Subject Classification: 58F08, 28A80.