Abstract: In many physical applications the equations describing a system are invariant under some transformation group. When bifurcation problems arise in such a situation, the group invariance may lead to multiplicities of the branch point. The main goal of the present paper is to demonstrate in a precise way the application of group representation theory to bifurcation theory. Group representation theory is a linear one, while bifurcation theory deals with the branch points of nonlinear unctional equations. Nevertheless, the theory of group representations applies to those nonlinear problems in a natural and elegant manner. The link between the two disciplines lies in the tensor character of the bifurcation equations on the one hand, and the theory of tensor products of group representations on the other.

Abstract: Let k be an algebraically closed field of characteristic p and G a finite group such thatp divides the order of G. We compute all almost split sequences over kG when kG is of finite representation type, or more generally, for a finite dimensional k-algebra A given by a Brauer tree. We apply this to show that if A and A' are stably equivalent k-algebras given by Brauer trees, then they have the same number of simple modules. Introduction. Let A be an artin algebra, that is, an artin ring which is finitely generated as a module over its center. Denote by mod A the category of finitely generated (left) A-modules. We recall from [5] that a nonsplit exact sequence 0 -O A 4 B -4 C -O 0 in mod A is said to be almost split if A and C are indecomposable, and given any morphism h: X -* C which is not a splittable epimorphism, there is a morphism s: X -* B such that gs = h. Given an indecomposable nonprojective C in mod A (or an indecomposable noninjective A in mod A), we have existence and uniqueness of an almost split sequence 0 -O A 4 B -> C -O0 [5]. A further study of the invariants determined by almost split sequences, for example the number of summands in a decomposition of the middle term B as a direct sum of indecomposable modules, is made in [6], and some methods for computing almost split sequences are discussed in [7]. Since the structure of almost split sequences is closely connected to the representation theory of the ring, it is of interest to obtain as much information as possible about them. In this paper we describe all almost split sequences for a particular class of finite dimensional k-algebras A, where k throughout this paper will denote an algebraically closed field; namely algebras given by Brauer trees (see [9]). We say that A is given by a Brauer tree if the following conditions hold: Let P1, . . .e, P denote the nonisomorphic indecomposable projective A-modules, and SI, . . . , Se the nonisomorphic simple modules, where S Pi/rPi. Here r denotes the radical of A. There is a tree with e edges, which are in one-one correspondence with the indecomposable projective, and hence also Received by the editors March 30, 1976. AMS (MOS) subject classifications (1970). Primary 16A26, 16A46, 16A64; Secondary 18G05. O American Mathematical Society 1977

Abstract: This paper advocates a new approach to the theory of finite group representa-tions by applying exclusively the method of commuting operators in quantum me-chanics. The basic problems of group representation theory such as the labeling of irreducible representations, the finding of characters, irreducible bases and matrix ele-ments and the CG coefficients et al are all simplified to the solving of the eigenfunc-tions of a certain complete set of commuting operators. This method has the advan-tage of being concise in theory and easily manageable in practice.

Abstract: This thesis, is concerned with the determination of the connected centralizers of semi-simple elements in a Chevalley group. To deal with this problem we shall use the recent work of R. Carter [6] and a new tool for the study of algebraic groups - the so called Brauer complex. This complex has been first defined by J. Humphreys [11] in the context of the modular representation theory of the finite Chevalley groups of universal type. Now, in our version, the Brauer complex can be also used for the ordinary representation theory of the finite Chevalley groups of adjoint type. For, Deligne and Lusztig in their fundamental work [9] have constructed for these groups certain families of irreducible complex representations whose degrees can be obtained if we know what subgroups of the finite Chevalley groups are the connected centralizers of semi-simple elements.

Abstract: The purpose of this paper is to prove a result on the number of irreducible projective representations of a finite group with respect to a given factor set and a group of linear characters acting on them. It includes a determination of the number of classes of projectively equivalent representations as well as a result of M. Osima on the classes of linearly equivalent representations. Osima proved his result by defining and calculating on irreducible projective Brauer characters, whereas the method used here is based on Brauer's well-known result for the linear modular representations and an induction argument on suitable coverings.