11 research outputs found
Jack vertex operators and realization of Jack functions
We give an iterative method to realize general Jack functions from Jack
functions of rectangular shapes. We first show some cases of Stanley's
conjecture on positivity of the Littlewood-Richardson coefficients, and then
use this method to give a new realization of Jack functions. We also show in
general that vectors of products of Jack vertex operators form a basis of
symmetric functions. In particular this gives a new proof of linear
independence for the rectangular and marked rectangular Jack vertex operators.
Thirdly a generalized Frobenius formula for Jack functions was given and was
used to give new evaluation of Dyson integrals and even powers of Vandermonde
determinant.Comment: Expanded versio
Determinantal Construction of Orthogonal Polynomials Associated with Root Systems
We consider semisimple triangular operators acting in the symmetric component
of the group algebra over the weight lattice of a root system. We present a
determinantal formula for the eigenbasis of such triangular operators. This
determinantal formula gives rise to an explicit construction of the Macdonald
polynomials and of the Heckman-Opdam generalized Jacobi polynomials.Comment: 28 page
Applications of Laplace-Beltrami operator for Jack polynomials
We use a new method to study the Laplace-Beltrami type operator on the Fock
space of symmetric functions, and as an example of our explicit computation we
show that the Jack symmetric functions are the only family of eigenvectors of
the differential operator. As applications of this explicit method we find a
combinatorial formula for Jack symmetric functions and the
Littlewood-Richardson coefficients in the Jack case. As further applications,
we obtain a new determinantal formula for Jack symmetric functions. We also
obtained a generalized raising operator formula for Jack symmetric functions,
and a formula for the explicit action of Virasoro operators. Special cases of
our formulas imply Mimachi-Yamada's result on Jack symmetric functions of
rectangular shapes, as well as the explicit formula for Jack functions of two
rows or two columns.Comment: 19 pages. Corrected new version with new formulas on Jack function
Dunkl Operators and Canonical Invariants of Reflection Groups
Using Dunkl operators, we introduce a continuous family of canonical
invariants of finite reflection groups. We verify that the elementary canonical
invariants of the symmetric group are deformations of the elementary symmetric
polynomials. We also compute the canonical invariants for all dihedral groups
as certain hypergeometric functions.Comment: v2: few corrections, some proofs and references added; v3: published
versio
Orthogonal polynomials of compact simple Lie groups
Recursive algebraic construction of two infinite families of polynomials in
variables is proposed as a uniform method applicable to every semisimple
Lie group of rank . Its result recognizes Chebyshev polynomials of the first
and second kind as the special case of the simple group of type . The
obtained not Laurent-type polynomials are proved to be equivalent to the
partial cases of the Macdonald symmetric polynomials. Basic relation between
the polynomials and their properties follow from the corresponding properties
of the orbit functions, namely the orthogonality and discretization. Recurrence
relations are shown for the Lie groups of types , , , ,
, , and together with lowest polynomials.Comment: 34 pages, some minor changes were done, to appear in IJMM
Dunkl Operators and Canonical Invariants of Reflection Groups
Using Dunkl operators, we introduce a continuous family of canonical invariants of finite reflection groups. We verify that the elementary canonical invariants of the symmetric group are deformations of the elementary symmetric polynomials. We also compute the canonical invariants for all dihedral groups as certain hypergeometric functions
Determinantal construction of orthogonal polynomials associated with root systems
Retrieved October 30, 2007 from http://lanl.arxiv.org/find/math/1/au:+Morse_J/0/1/0/all/0/1We consider semisimple triangular operators acting in the sym-
metric component of the group algebra over the weight lattice of a root sys-
tem. We present a determinantal formula for the eigenbasis of such triangular
operators. This determinantal formula gives rise to an explicit construction
of the Macdonald polynomials and of the Heckman-Opdam generalized Jacobi
polynomials