43 research outputs found
A solution of the quantum Knizhnik Zamolodchikov equation of type
We construct a solution of Cherednik's quantum Knizhnik Zamolodchikov
equation associated with the root system of type . This solution is given
in terms of a restriction of a -Jordan-Pochhammer integral. As its
applicaton, we give an explicit expression of a special case of the Macdonald
polynomial of the type. Finally we explain the connection with the
representation of the Hecke algebra.Comment: 18 pages, AMS-Latex, no figure
Collective Field Description of Spin Calogero-Sutherland Models
Using the collective field technique, we give the description of the spin
Calogero-Sutherland Model (CSM) in terms of free bosons. This approach can be
applicable for arbitrary coupling constant and provides the bosonized
Hamiltonian of the spin CSM. The boson Fock space can be identified with the
Hilbert space of the spin CSM in the large limit. We show that the
eigenstates corresponding to the Young diagram with a single row or column are
represented by the vertex operators. We also derive a dual description of the
Hamiltonian and comment on the construction of the general eigenstates.Comment: 14 pages, one figure, LaTeX, with minor correction
Difference system for Selberg correlation integrals
The Selberg correlation integrals are averages of the products
with respect to the Selberg
density. Our interest is in the case , , when this
corresponds to the -th moment of the corresponding characteristic
polynomial. We give the explicit form of a matrix linear
difference system in the variable which determines the average, and we
give the Gauss decomposition of the corresponding matrix.
For a positive integer the difference system can be used to efficiently
compute the power series defined by this average.Comment: 21 page
Free Boson Representation of -Vertex Operators and their Correlation Functions
A bosonization scheme of the -vertex operators of \uqa for arbitrary
level is obtained. They act as intertwiners among the highest weight modules
constructed in a bosonic Fock space. An integral formula is proposed for
-point functions and explicit calculation for two-point function is
presented.Comment: 22 pages, latex file, UT-618 (revised version
All the Exact Solutions of Generalized Calogero-Sutherland Models
A collective field method is extended to obtain all the explicit solutions of
the generalized Calogero-Sutherland models that are characterized by the roots
of all the classical groups, including the solutions corresponding to spinor
representations for and cases.Comment: Latex, 17 pages. Title and abstract slightly changed, plus minor
correction
Braid Structure and Raising-Lowering Operator Formalism in Sutherland Model
We algebraically construct the Fock space of the Sutherland model in terms of
the eigenstates of the pseudomomenta as basis vectors. For this purpose, we
derive the raising and lowering operators which increase and decrease
eigenvalues of pseudomomenta. The operators exchanging eigenvalues of two
pseudomomenta have been known. All the eigenstates are systematically produced
by starting from the ground state and multiplying these operators to it.Comment: 11 pages, Latex, no figure
A Selberg integral for the Lie algebra A_n
A new q-binomial theorem for Macdonald polynomials is employed to prove an
A_n analogue of the celebrated Selberg integral. This confirms the g=A_n case
of a conjecture by Mukhin and Varchenko concerning the existence of a Selberg
integral for every simple Lie algebra g.Comment: 32 page
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
The N=1 triplet vertex operator superalgebras
We introduce a new family of C_2-cofinite N=1 vertex operator superalgebras
SW(m), , which are natural super analogs of the triplet vertex
algebra family W(p), , important in logarithmic conformal field
theory. We classify irreducible SW(m)-modules and discuss logarithmic modules.
We also compute bosonic and fermionic formulas of irreducible SW(m) characters.
Finally, we contemplate possible connections between the category of
SW(m)-modules and the category of modules for the quantum group
U^{small}_q(sl_2), q=e^{\frac{2 \pi i}{2m+1}}, by focusing primarily on
properties of characters and the Zhu's algebra A(SW(m)). This paper is a
continuation of arXiv:0707.1857.Comment: 53 pages; v2: references added; v3: a few changes; v4: final version,
to appear in CM