2004 Fiscal Year Final Research Report Summary
Soliton equations and combinatories
| Project/Area Number |
15540165
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | KYOTO UNIVERSITY |
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Principal Investigator |
SHIOTA Takahiro Kyoto Univ., Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (20243008)
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| Project Period (FY) |
2003 – 2004
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| Keywords | KP hierarchy / BKP hierarchy / Calogero-Moser system / tau function / matrix model |
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| Research Abstract |
As is well-known, the motion of poles of a KP rational solution obeys the hierarchy of Calogero-Moser dynamical systems (and similarly for the trigonometric and elliptic solutions of KP hierarchy). This fact may get renewed interest after recent discovery by J.F.Van Diejen on the relation between the zeros of KdV wave function and the Ruijsenaars-Schneider system. Both the KP and the Calogero-Moser hierarchies allow generalizations with various internal symmetries from Lie theory point of view. We studied the pole motion of rational solution of the BKP and other hierarchies which can be handled explicitly. In particular, for BKP rational solution we obtained a matrix pair X,Y which can be regarded as the B-analogue of the Moser pair, and represented its tau function as the Pfaffian of time-dependent linear combination of Y and powers of X.
We also studied, jointly with A.Yu.Orlov, solutions of hypergeometric type to various soliton equations, and interpreted them as discrete analogues (ones with integrals replaced by sums) of various matrix models. For this we expanded the partition function of normal matrix model by Schur functions (hence rediscovered that the partition functions are Toda tau functions, and specialized the time variables in an appropriate way to obtain a different representation of it in terms of the sum over partitions, and interpreted the resulting formulae as discrete versions of various matrix models. This way we obtained discrete versions of normal, Hermite, and unitary matrix models.
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Research Products
(2 results)
