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Let M (n) be the algebra (both Lie and associative) of n × n matrices over C. Then M (n) inherits a Poisson structure from its dual using the bilinear form (x, y) = −tr xy. The Gl(n) adjoint orbits are the symplectic leaves and the algebra, P (n), of polynomial functions on M (n) is a Poisson algebra. In particular if f ∈ P (n) then there is a corresponding… (More)
Astract We study here the ring QS n of Quasi-Symmetric Functions in the variables x 1 , x 2 ,. .. , x n. F. Bergeron and C. Reutenauer  formulated a number of conjectures about this ring, in particular they conjectured that it is free over the ring Λ n of symmetric functions in x 1 , x 2 ,. .. , x n. We present here an algorithm that recursively… (More)
The use of topological ideas to explore various aspects of graph theory, and vice versa, is a fruitful area of research. There are links with other areas of mathematics, such as design theory and geometry, and increasingly with such areas as computer networks where symmetry is an important feature. Other books cover portions of the material here, but there… (More)
This paper gives a classification of parabolic subalgebras of simple Lie algebras over C that are complexifications of parabolic subalgebras of real forms for which Lynch's vanishing theorem for generalized Whittaker modules is non-vacuous. The paper also describes normal forms for the admissible characters in the sense of Lynch (at least in the quasi-split… (More)
Let X be an irreducible Hermitian symmetric space of non-compact type and rank r. Let p ∈ X and let K be the isotropy group of p in the group of biholomorphic transformations. Let S denote the symmetric algebra in the holomorphic tangent space to X at p. Then S is multiplicity free as a representation of K and the irreducible constituents are parametrized… (More)
In this paper we announce a qualitative description of an important class of closed n-dimensional submanifolds of the m-dimensional sphere, namely, those which locally minimize the n-area in the same way that geodesics minimize the arc length and are themselves locally n-spheres of constant radius r; those r that may appear are called admissible. It is… (More)
Associated with an m × n matrix with entries 0 or 1 are the m-vector of row sums and n-vector of column sums. In this article we study the set of all pairs of these row and column sums for fixed m and n. In particular, we give an algorithm for finding all such pairs for a given m and n.
We determine the Hilbert series of measures of entanglement for 4 qubits. Various techniques of constructive invariant theory are applied to prove the formula.
We define a polynomial measure of multiparticle entanglement which is scalable, i.e., which applies to any number of spin-1 2 particles. By evaluating it for three particle states, for eigenstates of the one dimensional Heisenberg antiferromagnet and on quantum error correcting code subspaces, we illustrate the extent to which it quantifies global… (More)