Planar Harmonic Polynomials of Type B

The hyperoctahedral group is the Weyl group of type B and is associated with a two-parameter family of differential-difference operators T_i, i=1,..,N (the dimension of the underlying Euclidean space). These operators are analogous to partial derivative operators. This paper finds all the polynomials in N variables which are annihilated by the sum of the squares (T_1)^2+(T_2)^2 and by all T_i for i>2 (harmonic). They are given explicitly in terms of a novel basis of polynomials, defined by generating functions. The harmonic polynomials can be used to find wave functions for the quantum many-body spin Calogero model.Comment: 17 pages, LaTe

Vector Polynomials and a Matrix Weight Associated to Dihedral Groups

The space of polynomials in two real variables with values in a 2-dimensional irreducible module of a dihedral group is studied as a standard module for Dunkl operators. The one-parameter case is considered (omitting the two-parameter case for even dihedral groups). The matrix weight function for the Gaussian form is found explicitly by solving a boundary value problem, and then computing the normalizing constant. An orthogonal basis for the homogeneous harmonic polynomials is constructed. The coefficients of these polynomials are found to be balanced terminating $_4F_3$-series

Symbolic integration with respect to the Haar measure on the unitary group

We present IntU package for Mathematica computer algebra system. The presented package performs a symbolic integration of polynomial functions over the unitary group with respect to unique normalized Haar measure. We describe a number of special cases which can be used to optimize the calculation speed for some classes of integrals. We also provide some examples of usage of the presented package.Comment: 7 pages, two columns, published version, software available at: https://github.com/iitis/Int

Symmetric and Antisymmetric Vector-valued Jack Polynomials

Polynomials with values in an irreducible module of the symmetric group can be given the structure of a module for the rational Cherednik algebra, called a standard module. This algebra has one free parameter and is generated by differential-difference ("Dunkl") operators, multiplication by coordinate functions and the group algebra. By specializing Griffeth's (arXiv:0707.0251) results for the G(r,p,n) setting, one obtains norm formulae for symmetric and antisymmetric polynomials in the standard module. Such polynomials of minimum degree have norms which involve hook-lengths and generalize the norm of the alternating polynomial.Comment: 22 pages, added remark about the Gordon-Stafford Theorem, corrected some typo