N. J. Wildberger

Sums of adjoint orbits

We investigate a natural generalization of the problem of the description of the eigenvalues of the sum of two Hermitian matrices both of whose eigenvalues are known. We describe more generally the convolution of the invariant probability measures supported on any two adjoint orbits of a compact Lie group. Our techniques utilize the convexity results of Guillemin and Sternberg and Kirwan on the one hand, and the character formulae of Weyl and Kirillov on the other. Applications to representation theory are discussed.

ON THE FOURIER TRANSFORM OF A COMPACT SEMISIMPLE LIE GROUP

We develop a concrete Fourier transform on a compact Lie group by means of a symbol calculus, or *-product, on each integral co-adjoint orbit. These *-products are constructed by means of a moment map defined for each irreducible representation. We derive integral formulae for these algebra structures and discuss the relationship between two naturally occurring inner products on them. A global Kirillov-type character is obtained for each irreducible representation. The case of SU (2) is treated in some detail, where some interesting connections with classical spherical trigonometry are obtained.

Minuscule posets from neighbourly graph sequences

We construct minuscule posets, an interesting family of posets arising in Lie theory, algebraic geometry and combinatorics, from sequences of vertices of a graph with particular neighbourly properties.

Strong hypergroups of order three

Abstract This paper investigates the question of when a finite hypergroup with three elements is strong , that is satisfies the condition that its dual signed hypergroup is actually a hypergroup. We classify hermitian hypergroups of order three by weight into two-dimensional families and show that the algebraic conditions arising from duality yield four interesting curves in the plane which bound the character values of strong and non-strong hypergroups. By analysing the relations between these curves we discover that the stratum of strong hypergroups is connected for all weights in the range [4,∞) except for the subinterval [5,5 1 16 ] where there are two components. For weight equal to 5 the second component degenerates to a single point, the Golden hypergroup.

Actions of Finite Hypergroups

This paper is concerned with actions of finite hypergroups on sets. After introducing the definitions in the first section, we use the notion of ‘maximal actions’ to characterise those hypergroups which arise from association schemes, introduce the natural sub-class of *-actions of a hypergroup and introduce a geometric condition for the existence of *-actions of a Hermitian hypergroup. Following an insightful suggestion of Eiichi Bannai we obtain an example of the surprising phenomenon of a 3-element hypergroup with infinitely many pairwise inequivalent irreducible *-actions.

Duality and Entropy for Finite Commutative Hypergroups and Fusion Rule Algebras

We develop a theory of harmonic analysis and duality for finite commutative hypergroups by considering somewhat more general objects called signed hypergroups. A notion of entropy is defined, and a Second Law of Thermodynamics is established. Applications to group theory and to the fusion rule algebras of conformal field theory are given.

Generalized hypergroups and orthogonal polynomials

We study in this paper a generalization of the notion of a discrete hypergroup with particular emphasis on the relation with systems of orthogonal polynomials. The concept of a locally compact hypergroup was introduced by Dunkl [8], Jewett [12] and Spector [25]. It generalizes convolution algebras of measures associated to groups as well as linearization formulae of classical families of orthogonal polynomials, and many results of harmonic analysis on locally compact abelian groups can be carried over to the case of commutative hypergroups; see Heyer [11], Litvinov [17], Ross [22], and references cited therein. Orthogonal polynomials have been studied in terms of hypergroups by Lasser [15] and Voit [31], see also the works of Connett and Schwartz [6] and Schwartz [23] where a similar spirit is observed. The special case of a discrete hypergroup, particularly in the commutative case, goes back earlier. In fact the ground-breaking paper of Frobenius

Diagonalization in compact Lie algebras and a new proof of a theorem of Kostant

We exhibit a simple algorithmic procedure to show that any element of a compact Lie algebra is conjugate to an element of a fixed maximal abelian subalgebra. An estimate of the convergence of the algorithm is obtained. As an application, we provide a new proof of Kostants theorem on the projection of orbits onto a maximal abelian subalgebra. 0 Let M E M(n, C) be a Hermitian matrix and consider the problem of diagonalizing M, that is, finding a unitary n x n matrix g such that g-1Mg is diagonal. This problem is essentially equivalent to that of finding the eigenvalues and eigenvectors of M. We propose an algorithm for solving this problem which utilizes the Lie algebra structure of 9, the n x n skew-Hermitian matrices, and the adjoint action of G, the n x n unitary group, on j. In fact our method applies generally to any compact connected Lie group G and its Lie algebra . Fix a maximal torus T C G with Lie algebra t C O and let (0.1) g = tE E Oa aEl+ be the decomposition of 9 into weight spaces under the adjoint action of T. Here 1+ is a set of positive roots and each space Oa is two-dimensional. Given Z E 9, we will write (0.2) z = zO + E za aET+ corresponding to (0.1). The idea is then to choose a E 1+ such that Za has maximum norm and then find g E G such that Ad(g)Z has no Oa component. This turns out to be essentially a problem in SU(2), which we can solve using only quadratic equations. If d(Z) denotes the distance from Z to the subspace Received by the editors November 29, 1990 and, in revised form, February 25, 1992. 1991 Mathematics Subject Classification. Primary 22E15; Secondary 58F05.

Distributive Lattices Defined for Representations of Rank Two Semisimple Lie Algebras

For a rank two root system and a pair of nonnegative integers, using only elementary combinatorics we construct two posets. The constructions are uniform across the root systems

A Rational Approach to Trigonometry

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