134,675 research outputs found
Extremal Trigonometrical and Power Polynomials of Several Variables
We consider the set of the power non-negative polynomials of several
variables and its subset that consists of polynomials which can be represented
as a sum of squares. It is shown in the classic work by D.Hilbert that it is a
proper subset. Both sets are convex. In our paper we have made an attempt to
work out a general approach to the investigation of the extremal elements of
these convex sets. We also consider the class of non-negative rational
functions. The article is based on the following methods: 1.We investigate
non-negative trigonometrical polynomials and then with the help of the Calderon
transformation we proceed to the power polynomials. 2.The way of constructing
support hyperplanes to the convex sets is given in the paper
Zonal polynomials via Stanley's coordinates and free cumulants
We study zonal characters which are defined as suitably normalized
coefficients in the expansion of zonal polynomials in terms of power-sum
symmetric functions. We show that the zonal characters, just like the
characters of the symmetric groups, admit a nice combinatorial description in
terms of Stanley's multirectangular coordinates of Young diagrams. We also
study the analogue of Kerov polynomials, namely we express the zonal characters
as polynomials in free cumulants and we give an explicit combinatorial
interpretation of their coefficients. In this way, we prove two recent
conjectures of Lassalle for Jack polynomials in the special case of zonal
polynomials.Comment: 45 pages, second version, important change
A probabilistic interpretation of the Macdonald polynomials
The two-parameter Macdonald polynomials are a central object of algebraic
combinatorics and representation theory. We give a Markov chain on partitions
of k with eigenfunctions the coefficients of the Macdonald polynomials when
expanded in the power sum polynomials. The Markov chain has stationary
distribution a new two-parameter family of measures on partitions, the inverse
of the Macdonald weight (rescaled). The uniform distribution on permutations
and the Ewens sampling formula are special cases. The Markov chain is a version
of the auxiliary variables algorithm of statistical physics. Properties of the
Macdonald polynomials allow a sharp analysis of the running time. In natural
cases, a bounded number of steps suffice for arbitrarily large k
- …