672 research outputs found
A generating function for the trace of the Iwahori-Hecke algebra
The Iwahori-Hecke algebra has a ``natural'' trace . This trace is the
evaluation at the identity element in the usual interpretation of the
Iwahori-Hecke algebra as a sub-algebra of the convolution algebra of a p-adic
semi-simple group. The Iwahori-Hecke algebra contains an important commutative
sub-algebra , that was described and studied by Bernstein,
Zelevinski and Lusztig. In this note we compute the generating function for the
value of on the basis
Intertwining operator for Calogero-Moser-Sutherland system
We consider generalised Calogero-Moser-Sutherland quantum Hamiltonian
associated with a configuration of vectors on the plane which is a union
of and root systems. The Hamiltonian depends on one parameter.
We find an intertwining operator between and the Calogero-Moser-Sutherland
Hamiltonian for the root system . This gives a quantum integral for of
order 6 in an explicit form thus establishing integrability of .Comment: 24 page
On the evaluation formula for Jack polynomials with prescribed symmetry
The Jack polynomials with prescribed symmetry are obtained from the
nonsymmetric polynomials via the operations of symmetrization,
antisymmetrization and normalization. After dividing out the corresponding
antisymmetric polynomial of smallest degree, a symmetric polynomial results. Of
interest in applications is the value of the latter polynomial when all the
variables are set equal. Dunkl has obtained this evaluation, making use of a
certain skew symmetric operator. We introduce a simpler operator for this
purpose, thereby obtaining a new derivation of the evaluation formula. An
expansion formula of a certain product in terms of Jack polynomials with
prescribed symmetry implied by the evaluation formula is used to derive a
generalization of a constant term identity due to Macdonald, Kadell and Kaneko.
Although we don't give the details in this work, the operator introduced here
can be defined for any reduced crystallographic root system, and used to
provide an evaluation formula for the corresponding Heckman-Opdam polynomials
with prescribed symmetry.Comment: 18 page
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