Physical Combinatorics

本课题的主要研究成果包括以下五个部分：(1)利用无穷远表示定义的次数，在共形协不变量空间中定义了滤子。我们得到了用Kostka多项式来表示伴随分次空间的特征的公式。在这个公式中，我们使用了融合乘积的协不变量的性质。所得公式为费米子型。利用Jack多项式和Macdonald多项式构造了满足一定零条件的对称多项式空间的基，称为轮条件。当二重仿射Hecke代数的多项式表示在参数的特定值是可约的时，这个空间就是它的极小子表示。(Iii)我们将表征可积量子场论形式因子的差分方程组的解空间实现为量子群表示理论中的无限维齐次空间。(Iv)我们利用基本场的傅立叶分量及其满足的二次关系构造了Virasoro代数的极小表示的单项基。(V)得到了量子自旋链的$n$点关联函数的$n$-折叠积分公式。通过求解qKZ差分方程，我们得到了一个不需要积分的代数公式。我们使用了带有连续维辅助空间的传递矩阵。在XYZ模型的情况下，我们需要Sklyanin代数中迹的表达式。我们证明了迹的计算归结为代数中的7个元素，并用椭圆函数来表示它们的迹。

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.