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)
2008 iii To my family.
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.
The main purpose of this article is to provide an alternate proof to a result of Perelman on gradient shrinking solitons. In dimension three we also generalize the result by removing the κ-non-collapsing assumption. In high dimension this new method allows us to prove a classification result on gradient shrinking solitons with vanishing Weyl curvature… (More)