Affine Weyl groups, discrete dynamical systems and Painleve equations

A new class of representations of affine Weyl groups on rational functions are constructed, in order to formulate discrete dynamical systems associated with affine root systems. As an application, some examples of difference and differential systems of Painleve type are discussed.Comment: AMSLaTeX, 16 page

Five-dimensional AGT Relation and the Deformed beta-ensemble

We discuss an analog of the AGT relation in five dimensions. We define a q-deformation of the beta-ensemble which satisfies q-W constraint. We also show a relation with the Nekrasov partition function of 5D SU(N) gauge theory with N_f=2N.Comment: 38page. References and an appendix for 4D case added. Typos correcte

Crystalline Spinon Basis for RSOS Models

The crystalline spinon basis for the RSOS models associated with $\widehat{sl_2}$ is studied. This basis gives fermionic type character formulas for the branching coefficients of the coset $(\widehat{sl_2})_l \times (\widehat{sl_2})_N/(\widehat{sl_2})_{l+N}$. In addition the path description of the parafermion characters is found as a limit of the spinon description of the string functions.Comment: 16 pages, latex, no figures, Comments are added in the discussion section. Some references are adde