Physical Combinatorics

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The main research results of the project consist of the following 5 parts.(i)We defined a filtration in the space of conformal coinvariants by using the degree defined by the representation at the infinity. We obtained a formula to represent the character of the associated gradedspace in terms of the Kostka polynomials. In this formula, we use the character of coinvariants for the fusion products. The obtained formulasare of fermionic type. We also obtained a bosonic formulas by solving a recursion relation.(ii)We constructed a basis of the space of symmetric polynomials satisfying a certain zero condition called the wheel condition by using the Jack and the Macdonald polynomials. This space is nothing but the minimal subrepresentation of the polynomial representation of the double affine Hecke algebra when it is reducible at the special values of the parameters.(iii)We realized the space of solutions to the system of difference equations which characterize the form factors of integrable quantum field theory, as an infinite dimensional homogeneous space in the representation theory of the quantum groups.(iv)We constructed a monomial basis of the minimal representations of the Virasoro algebra by using the Fourier components of the primary field and the quadratic relations satisfied by them.(v)Formulas in $n$-fold integrals for the $n$-point correlation functions of quantum spin chains are knwon. We obtained an algebraic formula without integrations by solving the qKZ difference equation. We used the transfer matrix with an auxiliary space of continuous dimensions. In the case of the XYZ model, we need the expression for the trace in Sklyanin's algebra. We proved that the calculation of the trace reduces to 7 elements in the algebra, and expressed thier traces by using elliptic theta functions.