Characterizing moore groups by tensor products of irreducible representations

It is shown that all irreducible representations of a σ-compact Lie group G have to be finite dimensional provided that for every π in the reduced dual of G the tensor product π ? \(\bar \pi \) has a discrete support.     

On the tensor product of representations of semisimple Lie groups

The problem of the decomposition of the tensor product of finite and infinite representations of a complex semigroup of a Lie group is examined by using the theory of characters of completely irreducible representations. A theorem is proved which indicates that completely irreducible representations enter into the expansion of the tensor product of a finite and elementary representation.Translated from Matematicheskie Zametki, Vol. 16, No. 5. pp. 731–739, November, 1974.     

On graphical regular representations of direct products of groups

A given group G may or may not have the property that there exists a graph X such that the automorphism group of X is regular, as a permutation group, and isomorphic to G. Mark E. Watkins has shown that the direct product of two finite groups has this property if each factor has this property and both factors are different from the cyclic group of order 2. Later, Wilfried Imrich generalized this result to infinite groups. In this paper, a new proof of this result for finite groups is given. The proof rests heavily on the result which states that if X is a graphical regular representation of the group G, then X is not self-complementary.     

Compactness in dual spaces of locally compact groups and tensor products of irreducible representations

We deal with supports of tensor products of irreducible representations of locally compact groups. The main result describes the structure of almost connected groups for which tensor products of representations in the reduced dual have compact supports. An immediate consequence is that an infinite almost connected group cannot have a compact reduced dual. Furthermore, the continuity problem for these supports is studied.     

Gaussian maps and tensor products of irreducible representations

Let G be a simply-connected complex simple Lie group, P a parabolic subgroup, L an ample line bundle on G/P. Then Γ(L) is an irreducible G-module, and every such arises in this way. In this paper, we initiate the study the G-filtration of the tensor product Γ(L)⊗Γ(M) by order of vanishing along the diagonal of G/P×G/P. Each subquotient of the filtration is contained in some Γ(L⊗M⊗SⁱΩ _G/P ¹ , so these (reducible) G-modules are examined. The first non-trivial piece of the filtration is given by a Gaussian map of [W3]; we conjecture the surjectivity in case L, M are ample, and prove this in a number of cases. One is led immediately to the study of a natural P-filtration of Γ(L) by order of vanishing of hyperplanes at a fixed point of G/P; tensoring this filtration with Γ(M) and inducing up to G gives the diagonal filtration, from which many familiar results are deduced.     

On tensor representations of knot algebras

Summary The fact that a Yang-Baxter operator defines tensor representations of the Artin braid group has been used to construct knot invariants. The main purpose of this note is to extend the tensor representations of the Artin braid group to representations of the braid groupZ B _k associated to the Coxeter graphB _k. This extension is based on some fundamental identities for the standardR-matrices of quantum Lie theory, here called four braid relations. As an application, tensor representations of knot algebras of typeB (Hecke, Temperley-Lieb, Birman-Wenzl-Murakami) are derived.     

The exponents of nonabelian tensor products of groups

We prove that the exponent of the nonabelian tensor product of two locally finite groups can be bounded in terms of exponents of given groups. Several estimates for the exponents of nonabelian tensor squares are obtained. In particular, if the group G is nilpotent of class ≤3 and of finite exponent, then the exponent of its nonabelian tensor square divides the exponent of G.     

Graphical regular representations of free products of groups

If X is a Cayley graph of a group G possessing a normal subgroup N, then there is a quotient graph of X which is a Cayley graph of $\text{G}\text{N}$. With the aid of this result, it is shown that the free product of at least two and at most countably many groups, each of which is at most countably generated, admits a graphical regular representation.     

On almost representations of groups

We say that a group belongs to the class if every nonunit quotient group of has an element of order two.

Let be a Hilbert space and let be its group of unitary operators. Suppose that groups and belong to the class and the order of is more than two. Then the free product has the following property. For any there exists a mapping satisfying the following conditions :

1)

2) for any representation the relation

holds.

     

Wreath products and representations of p-local finite groups

Given two finite p-local finite groups and a fusion preserving morphism between their Sylow subgroups, we study the question of extending it to a continuous map between their classifying spaces. The results depend on the construction of the wreath product of p-local finite groups which is also used to study p-local permutation representations.     

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).