We study centralizers of elements in domains. We generalize a result of the author and Small [4], showing that if A is a finitely generated noetherian domain and a ∈ A is not algebraic over the extended centre of A then the centralizer of a has Gelfand-Kirillov dimension at most one less than the Gelfand-Kirillov dimension of A. In the case that A is a finitely generated noetherian domain of GK...

We survey the known results on two old open problems on commutation of languages. The first problem, raised by Conway in 1971, is asking if the centralizer of a rational language must be rational as well – the centralizer of a language is the largest set of words commuting with that language. The second problem, proposed by Ratoandromanana in 1989, is asking for a characterization of those lang...

We introduce the linear centralizer method for a passive adversary to extract the shared key in group-theory based key exchange protocols (KEPs). We apply this method to obtain a polynomial time cryptanalysis of the Commutator KEP, introduced by Anshel–Anshel–Goldfeld in 1999 and considered extensively ever since. We also apply this method to the Centralizer KEP, introduced by Shpilrain–Ushakov...

We study a q-analog Qr(n, q) of the partition algebra Pr(n). The algebra Qr(n, q) arises as the centralizer algebra of the finite general linear group GLn(Fq) acting on a vector space IR q coming from r-iterations of Harish-Chandra restriction and induction. For n ≥ 2r, we show that Qr(n, q) has the same semisimple matrix structure as Pr(n). We compute the dimension dn,r(q) = dim(IR r q ) to be...

A modular theory for permutation representations and their centralizer rings is presented, analogous in several respects to the classical work of Brauer on group algebras. Some principal ingredients of the theory are characters of indecomposable components of the permutation module over a p-adic ring, modular characters of the centralizer ring, and the action of normalizers of p-subgroups P on ...

AND Abstract. Graded nilpotent Lie groups, or Carnot Groups are to subRiemannian geometry as Euclidean spaces are to Riemannian geometry. They are the metric tangent cones for this geometry. Hoping that the analogy between subRiemannian and Riemannian geometry is a strong one, one might conjecture that the subRiemannian geo-desic ow on any Carnot group is completely integrable. We prove this co...

Let (G;) be a permutation group of degree n. Let V (G;) be the set of all square matrices of order n which commute with all permutation matrices corresponding to permutations from (G;): V (G;) is a matrix algebra which is called the centralizer algebra of (G;): In this paper we introduce the combinatorial analogue of centralizer algebras, namely coherent (cellular) algebras and consider the pro...