| Budget Amount *help |
¥3,870,000 (Direct Cost: ¥3,600,000、Indirect Cost: ¥270,000)
Fiscal Year 2007: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2006: ¥900,000 (Direct Cost: ¥900,000)
Fiscal Year 2005: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2004: ¥1,000,000 (Direct Cost: ¥1,000,000)
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| Research Abstract |
The aim of this project is to study the correlation functions for Baxter's eight-vertex model, by applying the representation theory of the deformed W-algebras of type D. Here, the deformed W-algebras are certain variation of the so-called elliptic quantum groups, which are defined by specifying a set of elliptic functions (structure functions). I summarize my results in what follows.
(1) Using the deformed W-algebras, the vertex operators for Baxter's eight-vertex model are explicitly constructed. The rank of the W-algebra is determined by the arithmetic property of the so-called crossing parameter of the model.
(2) It is conjectured that the matrix elements of the vertex operators are uniquely characterized by a certain integral transformation, which commutes with the action of the Macdonald difference operators. We can regard the problem a discrete analogue of the initial value problem associated with integrable dynamical systems.
(3) To find a proof of the above conjecture, I studied
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the eigenvalue problem associated with the integral operator, and the Macdonald difference operators acting on the space of formal power series (not on the space of the symmetric polynomials). Not only the case of A-type root lattice, I treated the case of B, C, D, and the BC case, too. I found several quite nontrivial explicit formulas for such series by using computer algebra. Hence, I started to have communication with M. Noumi and found the kernel functions for the Koornwinder difference operator. As an application we found explicit formulas for the Koornwinder polynomials associated with single row and single column cases
(4) I studied the integrals of motion associated with the deformed W-algebras. By taking a classical limit, I derived an integrable hierarchy whish is described by the Sato theory with a certain reduction condition. The tau-function of this system satisfied a version of the Hirota-Miwa bilinear equation. In some degeneration limits, I found that a class of special solutions to the bilinear equation can be constructed, and the corresponding integrals of motion can be explicitly calculated. The results indicate that the classical mechanical object have a good amount of information which Macdonald polynomials or Hall-Littlewood polynomials have. Less
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