99 research outputs found
Laguerre and Meixner orthogonal bases in the algebra of symmetric functions
Analogs of Laguerre and Meixner orthogonal polynomials in the algebra of
symmetric functions are studied. This is a detailed exposition of part of the
results announced in arXiv:1009.2037. The work is motivated by a connection
with a model of infinite-dimensional Markov dynamics.Comment: Latex, 52p
Projections of orbital measures, Gelfand-Tsetlin polytopes, and splines
The unitary group U(N) acts by conjugations on the space H(N) of NxN
Hermitian matrices, and every orbit of this action carries a unique invariant
probability measure called an orbital measure. Consider the projection of the
space H(N) onto the real line assigning to an Hermitian matrix its (1,1)-entry.
Under this projection, the density of the pushforward of a generic orbital
measure is a spline function with N knots. This fact was pointed out by Andrei
Okounkov in 1996, and the goal of the paper is to propose a multidimensional
generalization. Namely, it turns out that if instead of the (1,1)-entry we cut
out the upper left matrix corner of arbitrary size KxK, where K=2,...,N-1, then
the pushforward of a generic orbital measure is still computable: its density
is given by a KxK determinant composed from one-dimensional splines. The result
can also be reformulated in terms of projections of the Gelfand-Tsetlin
polytopes.Comment: 12 pages; to appear in Journal of Lie Theor
- …