Seminar on Representation Theory Macdonald polynomials at roots of unity and GarsiaHaiman modules of the symmetric groups (in Japanese)
 Date

20060207 16:30 
20060207 18:00
 Place
 北大理学部3号館508室

Speaker/Organizer
 Hideaki MORITA

 This talk is partly based on a joint work with F. Descouens and J. Y. Thibon, Universite de MarnelaValle. 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 oneparametered symmetric polynomials, HallLittlewood polynomials. In 1993, LascouxLeclercThibon considered HallLitlewood 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 HallLittlewood polynomials do. It is known that the Macdonald polynomials give the graded characters of certain doubly graded modules of the symmetric group,called the GarsiaHaiman 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 GarsiaHaiman modules