Towards the Calculation of Jack Polynomials

<p>Jack polynomials are useful in mathematical statistics, but they are awkward to calculate, and their uses have chiefly been theoretical. In this thesis a determinantal expansion of Jack polynomials in elementary symmetric polynomials is found, complementing a recent result in the literature on expansions as determinants in monomial symmetric functions. These results offer enhanced possibilities for the calculation of these polynomials, and for finding workable approximations to them. The thesis investigates the structure of the determinants concerned, finding which terms can be expected to dominate, and quantifying the sparsity of the matrices involved. Expressions are found for the elementary and monomial symmetric polynomials when the variates involved assume the form of arithmetic and geometric progressions. The latter case in particular is expected to facilitate the construction of algorithms suitable for approximating Jack polynomials.</p>

Towards the Calculation of Jack Polynomials

<p>Jack polynomials are useful in mathematical statistics, but they are awkward to calculate, and their uses have chiefly been theoretical. In this thesis a determinantal expansion of Jack polynomials in elementary symmetric polynomials is found, complementing a recent result in the literature on expansions as determinants in monomial symmetric functions. These results offer enhanced possibilities for the calculation of these polynomials, and for finding workable approximations to them. The thesis investigates the structure of the determinants concerned, finding which terms can be expected to dominate, and quantifying the sparsity of the matrices involved. Expressions are found for the elementary and monomial symmetric polynomials when the variates involved assume the form of arithmetic and geometric progressions. The latter case in particular is expected to facilitate the construction of algorithms suitable for approximating Jack polynomials.</p>

Singular nonsymmetric Jack polynomials for some trapeziform tableaux

Let [Formula: see text] be symmetric group of degree [Formula: see text]. In the representation of [Formula: see text] corresponding to trapeziform partition (precisely [Formula: see text], [Formula: see text]), we propose nonsymmetric Jack polynomials in [Formula: see text] variables which are singular at [Formula: see text]. Moreover those nonsymmetric Jack polynomials span a module of [Formula: see text] which is isomorphic to the representation of type [Formula: see text].

New Pieri Type Formulas for Jack Polynomials and their Applications to Interpolation Jack Polynomials

We present new Pieri type formulas for Jack polynomials. As an application, we give a new derivation of higher order difference equations for interpolation Jack polynomials originally found by Knop and Sahi. We also propose Pieri formulas for interpolation Jack polynomials and intertwining relations for a kernel function for Jack polynomials.

Macdonald Polynomials and Chromatic Quasisymmetric Functions

We express the integral form Macdonald polynomials as a weighted sum of Shareshian and Wachs' chromatic quasisymmetric functions of certain graphs. Then we use known expansions of these chromatic quasisymmetric functions into Schur and power sum symmetric functions to provide Schur and power sum formulas for the integral form Macdonald polynomials. Since the (integral form) Jack polynomials are a specialization of integral form Macdonald polynomials, we obtain analogous formulas for Jack polynomials as corollaries.

Cumulants of Jack symmetric functions and b-conjecture (extended abstract)

International audience Goulden and Jackson (1996) introduced, using Jack symmetric functions, some multivariate generating series ψ(x, y, z; t, 1 + β) that might be interpreted as a continuous deformation of the rooted hypermap generating series. They made the following conjecture: coefficients of ψ(x, y, z; t, 1+β) are polynomials in β with nonnegative integer coefficients. We prove partially this conjecture, nowadays called b-conjecture, by showing that coefficients of ψ(x, y, z; t, 1 + β) are polynomials in β with rational coefficients. Until now, it was only known that they are rational functions of β. A key step of the proof is a strong factorization property of Jack polynomials when α → 0 that may be of independent interest.

Singular Nonsymmetric Jack Polynomials for Some Rectangular Tableaux

In the intersection of the theories of nonsymmetric Jack polynomials in N variables and representations of the symmetric groups S N one finds the singular polynomials. For certain values of the parameter κ there are Jack polynomials which span an irreducible S N -module and are annihilated by the Dunkl operators. The S N -module is labeled by a partition of N, called the isotype of the polynomials. In this paper the Jack polynomials are of the vector-valued type, i.e., elements of the tensor product of the scalar polynomials with the span of reverse standard Young tableaux of the shape of a fixed partition of N. In particular, this partition is of shape m , m , … , m with 2 k components and the constructed singular polynomials are of isotype m k , m k for the parameter κ = 1 / m + 2 . This paper contains the necessary background on nonsymmetric Jack polynomials and representation theory and explains the role of Jucys–Murphy elements in the construction. The main ingredient is the proof of uniqueness of certain spectral vectors, namely the list of eigenvalues of the Jack polynomials for the Cherednik–Dunkl operators, when specialized to κ = 1 / m + 2 . The paper finishes with a discussion of associated maps of modules of the rational Cherednik algebra and an example illustrating the difficulty of finding singular polynomials for arbitrary partitions.

Exact solutions by integrals of the non-stationary elliptic Calogero–Sutherland equation

We use generalized kernel functions to construct explicit solutions by integrals of the non-stationary Schrödinger equation for the Hamiltonian of the elliptic Calogero–Sutherland model (also known as elliptic Knizhnik–Zamolodchikov–Bernard equation). Our solutions provide integral representations of elliptic generalizations of the Jack polynomials.