Reflection group

Abstract: Any finite group of linear transformations on n variables leaves invariant a positive definite Hermitian form, and can therefore be expressed, after a suitable change of variables, as a group of unitary transformations (5, p. 257). Such a group may be thought of as a group of congruent transformations, keeping the origin fixed, in a unitary space Un of n dimensions, in which the points are specified by complex vectors with n components, and the distance between two points is the norm of the difference between their corresponding vectors.

Abstract: There is a theory of spherical harmonics for measures invariant under a finite reflection group. The measures are products of powers of linear functions, whose zero-sets are the mirrors of the reflections in the group, times the rotation-invariant measure on the unit sphere in Rn . A commutative set of differential-difference operators, each homogeneous of degree -1, is the analogue of the set of first-order partial derivatives in the ordinary theory of spherical harmonics. In the case of R2 and dihedral groups there are analogues of the Cauchy-Riemann equations which apply to Gegenbauer and Jacobi polynomial expansions. The analysis of orthogonality structures for polynomials in several variables is a problem of vast dimensions. This paper is part of an ongoing program to establish a workable theory for one particular class. The underlying structure is based on finite Coxeter groups: these are finite groups acting on Euclidean space, generated by reflections in the zero sets of a collection of linear functions (the "roots"); the weight functions for the orthogonality are products of powers of these linear functions restricted to the surface of the unit sphere. In addition, the weight function is required to be invariant under the action of the group. The resulting theory has strong similarities to the theory of spherical harmonics; this was established in previous papers of the author [3, 4, 5]. Most notably, a homogeneous polynomial is orthogonal to all polynomials of lower degree if and only if it is annihilated by a certain second-order differential-difference operator. Ordinary partial differentiation acts as an endomorphism on ordinary harmonic functions; the use of such operators and their adjoints leads to recurrence formulas and orthogonal decompositions for harmonic polynomials. In this paper we construct a commutative set of first-order differentialdifference operators associated to the second-order operator previously mentioned. Received by the editors June 1, 1987. 1980 Mathematics Subject Classification (1985 Revision). Primary 33A45, 33A65, 20H15; Secondary 20C30, 42C10, 51F15.

Abstract: Root systems and Coxeter groups are important tools in multivariable analysis. This paper is concerned with differential-difference and integral operators, and orthogonality structures for polynomials associated to Coxeter groups. For each such group, the structures allow as many parameters as the number of conjugacy classes of reflections. The classical orthogonal polynomials of Gegenbauer and Jacobi type appear in this theory as two-dimensional cases. For each Coxeter group and admissible choice of parameters there is a structure analogous to spherical harmonics which relies on the connection between a Laplacian operator and orthogonality on the unit sphere with respect to a group-invariant measure. The theory has been developed in several papers of the author [4,5,6,7]. In this paper, the emphasis is on the study of an intertwining operator which allows the transfer of certain results about ordinary harmonic polynomials to those associated to Coxeter groups. In particular, a formula and a bound are obtained for the Poisson kernel.

Abstract: This chapter is of an auxiliary nature and contains the modicum of the theory of finite reflection groups and Coxeter groups which we need for a systematic development of the theory of Coxeter matroids. A reflection group W is a finite subgroup of the orthogonal group of ℝ n generated by some reflections in hyperplanes (mirrors or walls). The mirrors cut ℝ n into open polyhedral cones, called chambers. The geometric concepts associated with the resulting chamber system (called the Coxeter complex of W) form the language of the theory of Coxeter matroids. The reader familiar with the theory of reflection groups and Coxeter groups may skip most of the chapter. However, we recommend that this reader look through Sections 5.12 “Residues,” 5.14 “Bruhat order” and 5.15 “Splitting the Bruhat order.”