49 research outputs found
Explicit formula for singular vectors of the Virasoro algebra with central charge less than 1
We calculate explicitly the singular vectors of the Virasoro algebra with the
central charge . As a result, we have an infinite sequence of the
singular vectors for each Fock space with given central charge and highest
weight, and all its elements can be written in terms of the Jack symmetric
functions with rectangular Young diagram.Comment: 10 pages, revised versio
Finding Rigged Configurations From Paths
We review reformulation of the map from tensor product of crystals to the
rigged configurations in terms of the energy function of affine crystals.
Especially, we give intuitive picture of the inverse scattering formalism for
the periodic box-ball systems formulated by Kuniba-Takagi-Takenouchi
(arXiv:math/0602481v2).Comment: 16 pages, accepted version for proceedings of ``Expansion of
Combinatorial Representation Theory" (RIMS, Kyoto University, October 2007
Bethe's Quantum Numbers And Rigged Configurations
We propose a method to determine the quantum numbers, which we call the
rigged configurations, for the solutions to the Bethe ansatz equations for the
spin-1/2 isotropic Heisenberg model under the periodic boundary condition. Our
method is based on the observation that the sums of Bethe's quantum numbers
within each string behave particularly nicely. We confirm our procedure for all
solutions for length 12 chain (totally 923 solutions).Comment: 16 pages. Supplementary tables are included in the source file. (v2)
New example at pages 8--9. (v3) Final version with minor revisio
Kirillov--Schilling--Shimozono bijection as energy functions of crystals
The Kirillov--Schilling--Shimozono (KSS) bijection appearing in theory of the
Fermionic formula gives an one to one correspondence between the set of
elements of tensor products of the Kirillov--Reshetikhin crystals (called
paths) and the set of rigged configurations. It is a generalization of
Kerov--Kirillov--Reshetikhin bijection and plays inverse scattering formalism
for the box-ball systems. In this paper, we give an algebraic reformulation of
the KSS map from the paths to rigged configurations, using the combinatorial R
and energy functions of crystals. It gives a characterization of the KSS
bijection as an intrinsic property of tensor products of crystals.Comment: 31 pages, final version, expositions much detaile