Infinite Analysis Seminar, Kyoto 2004

 

I. May 29, 2004, 11:00-12:15, RIMS, Room 206

DR Rei Inoue Yamazaki ( RIMS, Kyoto University )

TITLE:
Matrix realization of affine Jacobi varieties and the extended Lotka-Volterra lattice

ABSTRACT:   
Consider a gauge equivalence class M of polynomial matrices of a spectral parameter, and let X be the algebraic curve given by the common characteristic equation for M. It is known that M is isomorphic to the affine part of the Jacobi variety, Jaff(X).

In the context of classical integrable Hamiltonian systems, an orbit in M corresponds to the level set of some Lax matrix, and Jaff(X) is the invariant manifold of the system.

We construct a family of M and corresponded representatives explicitly by starting with certain types of Lax matrices, and study the isomorphic map from the representative to Jaff(X). Further we apply these to discuss the algebraic complete integrability of the extended Lotka-Volterra lattice with a periodic boundary condition.

This work corresponds to a partial extension of the result by Smirnov and Zeitlin.

 

  LUNCH: 12:20-13:30

 

II. May 29, 2004, 13:30-14:30, RIMS, Room 206

DR Masahiro Kasatani ( Kyoto University )

TITLE:
The vanishing ideal on double shifted diagonal and Jack and Macdonald polynomials

ABSTRACT:
We construct a ideal defined by certain zero condition on the double shifted diagonal. We give a basis by Jack and Macdonald polynomials and its linear combination with specialized parameters. A character formula (Hilbert-Poincaré series) is given.

Joint work with T.Miwa, A.N.Sergeev, A.P.Veselov


 

Research Institute for Mathematical Sciences, Kyoto University
Anatol Kirillov