A note on faithful representations of limit groups

In sec. 2, we have shown that all compact groups have faithful representations . For that purpose, the regular representations were constructed and examined. However, these last representations are infinite dimensional in general (i.e. when G is infinite). In particular, this does not prove the existence of irreducible representations (different from the identity in dimension 1) for compact groups. This section is devoted mainly to the proof of the following basic result (and its consequences). Theorem ( Peter-Weyl ). Let G be a compact group . For any s ≠ e in G, there exists a finite dimensional , irreducible representation π of G such that π(S) ≠ id. Since certain compact groups have no faithful finite dimensional representations (groups with arbitrarily small subgroups are in this class when infinite), this result is the best possible. This theorem is sometimes stated in the following terms: all compact groups have enough finite dimensional representations , or: all compact groups have a complete system of ( irreducible ) finite dimensional representations . As we have already seen that all finite dimensional representations of compact groups are completely reducible, the theorem will already be proved if we show that for s ≠ e in G, there exists a finite dimensional representation π with π(s) ≠ id. The proof of the preceding theorem is based on the spectral properties of compact hermitian operators in Hilbert spaces. Let us review the main points needed.

A minimal permutation representation of a group is its faithful permutation representation of least degree. Here the minimal permutation representations of finite simple exceptional twisted groups are studied: their degrees and point stabilizers, as well as ranks, subdegrees, and double stabilizers, are found. We can thus assert that, modulo the classification of finite simple groups, the aforesaid parameters are known for all finite simple groups.

A minimal permutation representation of a group is a faithful permutation representation of least degree. Well-studied to date are the minimal permutation representations of finite sporadic and classical groups for which degrees, point stabilizers, as well as ranks, subdegrees, and double stabilizers, have been found. Here we attempt to provide a similar account for finite simple ezceptional groups of types G2 and F4.

The present paper is the second part of [8] (brackets refer to bibliography) referred to below as Part I. We number sections, formulae, theorems, etc. consecutively from those in Part I, and use the same notation. For instance, f is a field of characteristic p with q (?_ cc ) elements; Zi is a t-vector space of dimension n; (5 = 65(n, f) is the full linear (modular) group of all f-automorphisms of 318 ; 93m is the space of all tensors of rank m over QI ; HIm is the Kronecker mth power representation of 6; 21m is the enveloping algebra of H1m. Superscripts zero indicate the analogous quantities defined over a field to of characteristic zero. The decomposition of non-modular tensors, or equivalently the determination of the reduced form of Ho is now a classical part of algebra. The objective of the present sequence of papers is to obtain a similar theory for modular tensors, or equivalently to study the reduced form of Hm . Any representation of 65 whose space is a subspace or factor space of 93m , or a direct sum of such spaces we call a tensor representation of 6. If q (and therefore 6) is finite we denote the group ring by r. One of the main results of the present paper is that there exists a faithful tensor representation of F (Th. X ?14). From this it follows from an unpublished theorem of Nesbitt that every representation of 6) is equivalent to a tensor representation, but we do not establish or apply this last result below. An important feature of the study of modular representations of groups has been the use of induced representations. One starts with a finite group and a non-modular representation of it, and after suitable number theoretic preparations take residue classes and obtain a modular representation. A generalization of this process would be to induce both the group and the representation. We leave for future investigation the determination of the general theory of such a process, and content ourselves here with the application of the idea to obtain from each irreducible representation of the non-modular full linear group a representation of the modular full linear group. This is done in ?9 below. In ?10 a character theory is developed for tensor representations of 6; this is applied in ?11 to prove that every irreducible representation of (M is equivalent to a tensor representation,1 and in ??12, 13 to obtain specific values of the irreducible and indecomposable modular characters for the representations H.m having m m. In ??15-20 the situation for m < 2p is cleaned up by re-

We describe a general construction of irreducible unitary representations of the group of currents with values in the semidirect product of a locally compact subgroup P 0 by a one-parameter group ℝ*+ = {r: r > 0} of automorphisms of P 0. This construction is determined by a faithful unitary representation of P 0 (canonical representation) whose images under the action of the group of automorphisms tend to the identity representation as r → 0. We apply this construction to the current groups of maximal parabolic subgroups in the groups of motions of the n-dimensional real and complex Lobachevsky spaces. The obtained representations of the current groups of parabolic subgroups uniquely extend to the groups of currents with values in the groups O(n, 1) and U(n, 1). This gives a new description of the representations, constructed in the 1970s and realized in the Fock space, of the current groups of the latter groups. The key role in our construction is played by the so-called special representation of the parabolic subgroup P and a remarkable σ-finite measure (Lebesgue measure) ℒ on the space of distributions.

Let G be any analytic group, let T be a maximal toroid of the radical of G, and let S be a maximal sernisimnple analytic subgroup of G. If L = L(G) is the Lie algebra of G, rad[L, L] is the radical of [L, L], and Z(L) is the center of L, we show that G has a faithful representation if and only if (i) rad[L, L] n Z(L) n L(T) = (0), and (ii) S has a faithful representation. A theorem of M. Moskowitz [4, Thm. 2], shows that if L is a finite-dimensional (real) Lie algebra, then all analytic groups with Lie algebra L have faithful representations if and only if (i) rad[L, L] n Z(L) = (0), and (ii) for some maximal semisimple subalgebra S of L, the simply connected analytic group with Lie algebra S has a faithful representation. So it would be of interest to find a similar criterion for a single analytic group G to have a faithful representation. Such a criterion is given in Theorem 2 below. As a consequence, we obtain Moskowitz' Theorem in Corollary 3. So our criterion in the solvable case says that G has a faithful representation if and only if [L, L] n Z(L) n L(T) = (0) for some maximal toroid T of G where L = L(G); whereas the well-known criterion in the solvable case is that G has a faithful representation if and only if [G, G] is closed in G and simply connected [2, p. 220]. For the case of semisimple analytic groups, we refer the reader to [2, pp. 199-201]. Our proof uses the notion of nuclei of analytic groups introduced by Hochschild and Mostow. A nucleus K of an analytic group G is a closed normal simply connected solvable analytic subgroup of G such that G/K is reductive. An analytic group G has a faithful representation if and only if G has a nucleus; if G has a nucleus K, then G = K P (semi-direct) for every maximal reductive analytic subgroup P of G [3, Section 2]. Recall that an analytic group is reductive if it has a faithful representation and all its representations are semisimple. If G is an analytic group, L:(G) is its Lie algebra, rad G is its radical, and [G, G] is its commutator (derived) subgroup. Similarly, if L is a Lie algebra, rad L is its radical, and [L, L] is its commutator (derived) subalgebra. All representations of analytic groups are assumed to be analytic and finite dimensional. Received by the editors October 26, 1995 and, in revised form, March 29, 1996. 1991 Mathematics Subject Classification. Primary 22E15, 22E60. ( 1997 American Mathematical Society

Some well-known classical results related to the description of integral representations of finite groups over Dedekind rings R, especially for the rings of integers Z and p-adic integers Zp and maximal orders of local fields and fields of algebraic numbers go back to classical papers by S. S. Ryshkov, P. M. Gudivok, A. V. Roiter, A. V. Yakovlev, W. Plesken. For giving an explicit description it is important to find matrix realizations of the representations, and one of the possible approaches is to describe maximal finite subgroups of GLn(R) over Dedekind rings R for a fixed positive integer n. The basic idea underlying a geometric approach was given in Ryshkov’s papers on the computation of the finite subgroups of GLn(Z) and further works by W. Plesken and M. Pohst. However, it was not clear, what happens under the extension of the Dedekind rings R in general, and in what way the representations of arbitrary p-groups, supersolvable groups or groups of a given nilpotency class can be approached. In the present paper the above classes of groups are treated, in particular, it is proven that for a fixed n and any given nonabelian p-group G there is an infinite number of pairwise non-isomorphic absolutely irreducible representations of the group G. A combinatorial construction of the series of these representations is given explicitly. In the present paper an infinite series of integral pairwise inequivalent absolutely irreducible representations of finite p-groups with the extra congruence conditions is constructed. We consider certain related questions including the embedding problem in Galois theory for local faithful primitive representations of supersolvable groups and integral representations arising from elliptic curves.

In this paper we construct a faithful representation of the mapping class group of the genus two surface into a group of matrices over the complex numbers. Our starting point is the Lawrence-Krammer representation of the braid group B_n, which was shown to be faithful by Bigelow and Krammer. We obtain a faithful representation of the mapping class group of the n-punctured sphere by using the close relationship between this group and B_{n-1}. We then extend this to a faithful representation of the mapping class group of the genus two surface, using Birman and Hilden's result that this group is a Z_2 central extension of the mapping class group of the 6-punctured sphere. The resulting representation has dimension sixty-four and will be described explicitly. In closing we will remark on subgroups of mapping class groups which can be shown to be linear using similar techniques.

Homotopy representations of [formula omitted] and Spin(7) at the prime 2

Soft set theory originated by Molodtsov is an effective technique for dealing with uncertainties. In the framework of soft set theory, soft group is a key concept of algebraic theory of soft sets. The aim of this paper is provided an initial study of the linear representations of soft groups. The fundamental notions such as linear and matrix representations, equivalent representation, faithful representation, trivial representation of soft groups together with some illustrative examples are introduced. The concepts of soft invariant space, irreducible representation, completely reducible representation are proposed. And then the relationships among these concepts are demonstrated by means of several related theorems.

This paper is an exploration of the faithful transitive permutation representations of the orientation-preserving automorphisms groups of orientably-regular toroidal maps and hypermaps. The main theorems of this paper list all possible degrees of these specific groups. This extends prior accomplishments of the authors, wherein their focus was confined to the study of the automorphisms groups of toroidal regular maps and hypermaps. In addition, the authors bring out the recently developed GAP package corefreesub that can be used to find faithful transitive permutation representations of any group. With the aid of this powerful tool, the authors show how Schreier coset graphs of the automorphism groups of toroidal maps and hypermaps can be easily constructed.

We define and study extensions of the Artin and Perron–Vannier representations of braid groups to topological and algebraic generalizations of braid groups. We provide faithful representations of braid groups of oriented surfaces with boundary as automorphisms of finitely generated free groups. The induced representations of such groups as outer automorphisms of finitely generated free groups are still faithful. Also we give a representation of braid groups of closed surfaces as outer automorphisms of finitely generated free groups. Finally, we provide faithful representations of Artin–Tits groups of type 𝒟 as automorphisms of free groups.

Cohomology of classifying spaces of central quotients of rank two Kac-Moody groups

In the paper it is proved, in particular, that a group is polycyclic if and only if it is soluble of finite rank, satisfies the ascending chain condition for normal subgroups and admits a faithful irreducible primitive representation over a field of characteristic zero. Methods are developed that enable one to study induced representations of nilpotent and soluble groups of finite rank.

In this paper, faithful matrix representations of the jet groups \(G^3_n\) are presented, following a detailed description of their components in block form. Such groups can be used further to study symmetries of differential equations. Elements of these matrix representations are derived.