Seminar on Representation Theory Macdonald polynomials at roots of unity and Garsia-Haiman modules of the symmetric groups (in Japanese)

2006-02-07 16:30 - 2006-02-07 18:00
Hideaki MORITA
This talk is partly based on a joint work with F. Descouens and J. -Y. Thibon, Universite de Marne-la-Valle. Macdonald polynomials are symmetric functions with two parameters $q$, $t$, introduced by I. G. Macdonald in 1988.These polynomials are generalization of a family of one-parametered symmetric polynomials, Hall-Littlewood polynomials. In 1993, Lascoux-Leclerc-Thibon considered Hall-Litlewood polynomials at roots of unity, and showed that they have nice properties, so called the "factorization formula" and the "plethystic formula". In this talk, we shall consider Macdonald polynomials at roots of unity, and see that they have similar nice properties

as Hall-Littlewood polynomials do. It is known that the Macdonald polynomials give the graded characters of certain doubly graded modules of the symmetric group,called the Garsia-Haiman modules. This work is motivated by a problem to understand a certain curious property of these modules, which we call in this talk as the "coincidence of dimension". We shall also see in this talk that how Macdonald polynomials at roots of unity relate the coincidence of dimension of Garsia-Haiman modules