Regular orbits on the deleted permutation module

Let S _d denote the symmetric group of degree d. Let F be the field of r elements, and let S(d, r) = S _d × F*. Let V(d, r) be the deleted permutation module of dimension d – 1 over F, viewed as an S(d, r)-module. We determine the pairs (d, r) such that S(d, r) has a regular orbit on V (d, r). This question arose from D. Goodwin’s work on the quasisimple case of the k(GV)-problem. Received: 17 April 2007     

Regular orbits of self-dual modules

We obtain a result regarding the existence of regular orbits for self-dual modules, thus generalizing a theorem of A. Espuelas for symplectic modules.

     

Regular orbits and positive directions

Let A be a bounded linear operator defined on a separable Banach space X. Then A is said to be supercyclic if there exists a vector x ∈ X (later called supercyclic for A), such that the projective orbit is dense in X. On the other hand, A is said to be positive supercyclic if for each supercyclic vector x, the positive projective orbit, is dense in X. Sometimes supercyclicity and positive supercyclicity are equivalent. The study of this relationship was initiated in [14] by F. León and V. Müller. In this paper we study positive supercyclicity for operators A of the form , with , defined on . We will see that such a problem is related with the study of regular orbits. The notion of positive directions will be central throughout the paper.      

On permutation representations of polyhedral groups

We answer affirmatively the following question of Derek Holt: Given integers , can one, in a simple manner, find a finite set and permutations such that has order , has order and has order ? The method of proof enables us to prove more general results (Theorems 2 and 3).

     

Regular orbits of hall π-subgroups

partially supported by DGICYT     

Abelian permutation groups with graphical representations

Journal of Algebraic Combinatorics - In this paper we characterize those automorphism groups of colored graphs and digraphs that are abelian as abstract groups. This is done in terms of basic...     

Deformations of permutation representations of Coxeter groups

The permutation representation afforded by a Coxeter group W acting on the cosets of a standard parabolic subgroup inherits many nice properties from W such as a shellable Bruhat order and a flat deformation over ?[q] to a representation of the corresponding Hecke algebra. In this paper we define a larger class of “quasiparabolic” subgroups (more generally, quasiparabolic W-sets), and show that they also inherit these properties. Our motivating example is the action of the symmetric group on fixed-point-free involutions by conjugation.     

Regular representations of discrete groups

Kaniuth [3] has given necessary and sufficient conditions for the regular representation of a discrete group to be type I or type II. We give a non-measure-theoretic proof of his results, plus additional information on the structure of the W^∗ algebra generated by the regular representation.     

Hopf subalgebras and tensor powers of generalized permutation modules

By means of a certain module V$V$ and its tensor powers in a finite tensor category, we study a question of whether the depth of a Hopf subalgebra R$R$ of a finite-dimensional Hopf algebra H$H$ is finite. The module V$V$ is the counit representation induced from R$R$ to H$H$, which is then a generalized permutation module, as well as a module coalgebra. We show that if in the subalgebra pair either Hopf algebra has finite representation type, or V$V$ is either semisimple with R^∗$R^{\ast }$ pointed, projective, or its tensor powers satisfy a Burnside ring formula over a finite set of Hopf subalgebras including R$R$, then the depth of R$R$ in H$H$ is finite. One assigns a nonnegative integer depth to V$V$, or any other H$H$-module, by comparing the truncated tensor algebras of V$V$ in a finite tensor category and so obtains upper and lower bounds for depth of a Hopf subalgebra. For example, a relative Hopf restricted module has depth 1, and a permutation module of a corefree subgroup has depth less than the number of values of its character.     

On product representations of powers, I

The solvability of the equation a₁a₂ … a_k = x², a₁, a₂, …, a_k ε is studied for fixed k and ‘dense’ sets of positive integers. In particular, it is shown that if k is even and k 4, and is of positive upper density, then this equation can be solved.     

Regular incidence permutation sets and incidence quasigroups

An incidence group (P,R·) can be considered as an incidence space (P,R) together with a groupC(=P set of left translations) of collineations of (P,R) acting regularly and fixed point free onP. Here we replaceC by a regular collineation set with fixed points. Then (P,R,C) can be turned only in an incidence quasigroup. Examples of such structures can be derived from arbitrary absolute spaces (P,R,). If denotes the set of all reflectionsp in pointsp ofP then is a set of collineations, even of dilatations, of the incidence space (P,R) acting regularly on the setP of points. Therefore we will study more precisely the case whereC consists of weak dilatations (§2) and then apply the results on geometric structures like kinematic spaces, affine geometries (§5) and hyperbolic planes (§6). By that we present new properties of these generalized dilatations, which were introduced by G. Kist and B. Reinmidl.This research was partially supported by M.U.R.S.T. (40% grant) and by G.N.S.A.G.A. of Italian C.N.R. while the first Author was Visiting Professor under a C.N.R. grant.Dedicated to Günter Pickert on the occasion of his 80th birthday