1,168 research outputs found

### 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

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