Matrix coefficient

Abstract: The compact matrix pseudogroup is a non-commutative compact space endowed with a group structure. The precise definition is given and a number of examples is presented. Among them we have compact group of matrices, duals of discrete groups and twisted (deformed)SU(N) groups. The representation theory is developed. It turns out that the tensor product of representations depends essentially on their order. The existence and the uniqueness of the Haar measure is proved and the orthonormality relations for matrix elements of irreducible representations are derived. The form of these relations differs from that in the group case. This is due to the fact that the Haar measure on pseudogroups is not central in general. The corresponding modular properties are discussed. The Haar measures on the twistedSU(2) group and on the finite matrix pseudogroup are found.

Abstract: Group Representations and Character Theory. Introduction to Modular Representations. Integral Representations: Orders and Lattices. Local and Global Theory of Integral Representations. Bibliography. Index.

Abstract: Hypergeometric functions of matrix argument arise in a diverse range of applications in harmonic analysis, multivariate statistics, quantum physics, molecular chemistry, and number theory. This paper presents a general theory of such functions for real division algebras. These functions, which generalize the classical hypergeometric functions, are defined by infinite series on the space S = S(n, F) of all n x n Hermitian matrices over the division algebra F. The theory depends intrinsically upon the representation theory of the general linear group G = GL(n, F) of invertible n x n matrices over F, and the theme of this work is the full exploitation of the inherent group theory. The main technique is the use of the method of "algebraic induction" to realize explicitly the appropriate representations of G, to decompose the space of polynomial functions on S, and to describe the "zonal polynomials" from which the hypergeometric functions are constructed. Detailed descriptions of the convergence properties of the series expansions are given, and integral representations are provided. Future papers in this series will develop the fine structure of these functions. 0. Introduction. We begin a series of articles in which we develop the fine structure of generalized hypergeometric functions of matrix argument. By "fine structure", we allude to the analogues of such classical results as series expansions, integral formulas, asymptotics, differential equations, summation formulas, addition theorems, composition formulas, and recurrence relations. This first paper, in which we simultaneously treat real, complex, and quaternionic analysis, is the result of our desire to present a complete theory of hypergeometric functions of matrix argument over real division algebras, not only as a framework for the body of detailed results to follow in later papers, but also to clarify the representationtheoretic foundation for such a theory. Although these hypergeometric functions are of interest on purely analytic grounds, they arise in a wide range of applications. Indeed, various classes of Received by the editors May 19, 1986. 1980 Malhema1zics S*ject Claszficain (1985 Ren). Primary 22E30, 22E45, 33A75, 43A85, 43A90, 62H10; Secondary 20G20, 32A07, 32M15, 44A10, 62E15. Key w and phrases. Generalized hypergeometric functions, zonal polynomials, representation theory, algebraic induction, multivariate statistics, general linear group, generalized gamma functions, Pochhammer symbols, Laplace transforms, maximal compact subgroup, invariant polynomials, positive cones, symmetric spaces, Schur functions, special functions of matrix argument. The first author is on leave from the University of Wyoming. The second author was partially supported by the National Science Foundation under Grant MCS-8403381, and by the Research Council of the University of North Carolina. (B1987 American Mathematical Society 0002-9947/87 $1.00 + $.25 per page

Abstract: Part 1. The concept of a vector space irreducible under a set of operators is developed from first principles, and then introduced into the many-particle formalism of quantum mechanics by means of an explicit postulate of irreducibility. The calculus of the irreducible matrix representations of finite groups is developed ab initio, including the theory of characters and projection operators. The use of this calculus to simplify eigenvalue calculations is explained in detail. Part 2. The structure of the general crystal space group, including glide-planes and screw-axes, is discussed briefly. The theory is developed of the symmetry group of a many-electron system with spin-orbit coupling, using the Dirac formalism. A detailed discussion is given of Wigner's time-reversal theorems for a many-electron system, including the character tests for time-reversal degeneracy. A general theory of the permutation symmetry of a many-electron system is developed, and shown to contain the Dirac vector model as a special case. A new treatment is given of the theory of the irreducible representations of space groups, including the double space groups and Herring's time-reversal theorems.