5 research outputs found
Powers of the Vandermonde determinant, Schur Functions, and recursive formulas
Since every even power of the Vandermonde determinant is a symmetric
polynomial, we want to understand its decomposition in terms of the basis of
Schur functions. We investigate several combinatorial properties of the
coefficients in the decomposition. In particular, we give recursive formulas
for the coefficient of the Schur function s_{\m} in the decomposition of an
even power of the Vandermonde determinant in variables in terms of the
coefficient of the Schur function s_{\l} in the decomposition of the same
even power of the Vandermonde determinant in variables if the Young diagram
of \m is obtained from the Young diagram of \l by adding a tetris type
shape to the top or to the left. An extended abstract containing the statement
of the results presented here appeared in the Proceedings of FPSAC11Comment: 23 pages; extended abstract appeared in the Proceedings of FPSAC1
Highest weight Macdonald and Jack Polynomials
Fractional quantum Hall states of particles in the lowest Landau levels are
described by multivariate polynomials. The incompressible liquid states when
described on a sphere are fully invariant under the rotation group. Excited
quasiparticle/quasihole states are member of multiplets under the rotation
group and generically there is a nontrivial highest weight member of the
multiplet from which all states can be constructed. Some of the trial states
proposed in the literature belong to classical families of symmetric
polynomials. In this paper we study Macdonald and Jack polynomials that are
highest weight states. For Macdonald polynomials it is a (q,t)-deformation of
the raising angular momentum operator that defines the highest weight
condition. By specialization of the parameters we obtain a classification of
the highest weight Jack polynomials. Our results are valid in the case of
staircase and rectangular partition indexing the polynomials.Comment: 17 pages, published versio
The Partition Function of Multicomponent Log-Gases
We give an expression for the partition function of a one-dimensional log-gas
comprised of particles of (possibly) different integer charge at inverse
temperature {\beta} = 1 (restricted to the line in the presence of a
neutralizing field) in terms of the Berezin integral of an associated non-
homogeneous alternating tensor. This is the analog of the de Bruijn integral
identities [3] (for {\beta} = 1 and {\beta} = 4) ensembles extended to
multicomponent ensembles.Comment: 14 page