Abstract. Fermionic formulae originate in the Bethe ansatz in solvable lattice models. They are specific expressions of some q-polynomials as sums of products of q-binomial coefficients. We consider the fermionic formulae associated with general non-twisted quantum affine algebra Uq(X (1) n ) and discuss several aspects related to representation theories and combinatorics. They include crystal base theory, one dimensional sums, spinon character formulae, Q-system and combinatorial completeness of the string hypothesis for arbitrary Xn.

Recently Witten proposed to consider elliptic genus in N = 2 superconformal field theory to understand the relation between N = 2 minimal models and Landau-Ginzburg theories. In this paper we first discuss the basic properties satisfied by elliptic genera in N = 2 theories. These properties are confirmed by some fundamental class of examples.Then we introduce a generic procedure to compute the elliptic genera of a particular class of orbifold theories, i.e. the ones orbifoldized by e 2πiJ 0 in the Neveu-Schwarz sector.This enables us to calculate the elliptic genera for Landau-Ginzburg orbifolds. When the Landau-Ginzburg orbifolds allow an interpretation as target manifolds with SU (N ) holonomy we can compare the expressions with the ones obtained by orbifoldizing tensor products of N = 2 minimal models. We also give sigma model expressions of the elliptic genera for manifolds of SU (N ) holonomy.

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In this paper a comprehensive review is given on the current status of achievements in the geometric aspects of the Painlevé equations, with a particular emphasis on the discrete Painlevé equations. The theory is controlled by the geometry of certain rational surfaces called the spaces of initial values, which are characterized by eight point configuration on P 1 × P 1 and classified according to the degeneration of points. We give a systematic description of the equations and their various properties, such as affine Weyl group symmetries, hypergeometric solutions and Lax pairs under this framework, by using the language of Picard lattice and root systems. We also provide with a collection of basic data; equations, point configurations/root data, Weyl group representations, Lax pairs, and hypergeometric solutions of all possible cases.

The relation between the charge of Lascoux-Schützenberger and the energy function in solvable lattice models is clarified. As an application, A. N. Kirillov's conjecture on the expression of the branching coefficient of sln/sln as a limit of Kostka polynomials is proved.Mathematics Subject Classification (1991). 81R50, 82B23, 17B37, 05A30.

Abstract. The fourth Painlevé equation PIV is known to have symmetry of the affine Weyl group of type A (1) 2 with respect to the Bäcklund transformations. We introduce a new representation of PIV , called the symmetric form, by taking the three fundamental invariant divisors as the dependent variables. A complete description of the symmetry of PIV is given in terms of this representation. Through the symmetric form, it turns out that PIV is obtained as a similarity reduction of the 3-reduced modified KP hierarchy. It is proved in particular that the special polynomials for rational solutions PIV , called Okamoto polynomials, are expressible in terms of the 3-reduced Schur functions.It is known by K. Okamoto [7] that the fourth Painlevé equation has symmetries under the affine Weyl group of type A 2 -symmetries become clearly visible. By means of this representation, we clarify the internal relation between the fourth Painlevé equation and the modified KP hierarchy. We obtain in particular a complete description of the rational solutions of the fourth Painlevé equation in terms of Schur functions. This implies that the so-called Okamoto polynomials, which arise from the τ -functions for rational solutions, are in fact expressible by the 3-reduced Schur functions. 1 §1. A symmetric form of the fourth Painlevé equationThe fourth Painlevé equation P IV is the following second order ordinary

We study an analog of the AGT (Alday-Gaiotto-Tachikawa) relation in five dimensions. We conjecture that the instanton partition function of 5D N = 1 pure SU (2) gauge theory coincides with the inner product of the Gaiotto-like state in the deformed Virasoro algebra. In four-dimensional case, a relation between the Gaiotto construction and the theory of Braverman and Etingof is also discussed.

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 Painlevé type are discussed.