80 research outputs found
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 -series
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
Some Singular Vector-valued Jack and Macdonald Polynomials
For each partition of there are irreducible modules of the
symmetric groups or the corresponding Hecke algebra
whose bases consist of reverse standard Young
tableaux of shape . There are associated spaces of nonsymmetric Jack and
Macdonald polynomials taking values in these modules, respectively.The Jack
polynomials are a special case of those constructed by Griffeth for the
infinite family of complex reflection groups. The
Macdonald polynomials were constructed by Luque and the author. For both the
group and the Hecke algebra
there is a commutative set of Dunkl operators. The Jack and the Macdonald
polynomials are parametrized by and respectively.
For certain values of the parameters (called singular values) there are
polynomials annihilated by each Dunkl operator; these are called singular
polynomials. This paper analyzes the singular polynomials whose leading term is
, where is an arbitrary reverse standard Young tableau
of shape . The singular values depend on properties of the edge of the
Ferrers diagram of .Comment: 19 page
Polynomials Associated with Dihedral Groups
There is a commutative algebra of differential-difference operators, with two
parameters, associated to any dihedral group with an even number of
reflections. The intertwining operator relates this algebra to the algebra of
partial derivatives. This paper presents an explicit form of the action of the
intertwining operator on polynomials by use of harmonic and Jacobi polynomials.
The last section of the paper deals with parameter values for which the
formulae have singularities.Comment: Published in SIGMA (Symmetry, Integrability and Geometry: Methods and
Applications) at http://www.emis.de/journals/SIGMA
Some Orthogonal Polynomials in Four Variables
The symmetric group on 4 letters has the reflection group as an
isomorphic image. This fact follows from the coincidence of the root systems
and . The isomorphism is used to construct an orthogonal basis
of polynomials of 4 variables with 2 parameters. There is an associated quantum
Calogero-Sutherland model of 4 identical particles on the line.Comment: This is a contribution to the Special Issue on Dunkl Operators and
Related Topics, published in SIGMA (Symmetry, Integrability and Geometry:
Methods and Applications) at http://www.emis.de/journals/SIGMA
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