Shi, Jian-Yi (1984) The decomposition into cells of the affine Weyl groups of type A. PhD thesis, University of Warwick.

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Abstract

In [1], Kazhdan and Lusztig introduce the concept of a W-graph for a Coxeter group W. In particular, they define left, right and two-sided cells. These W-graphs play an important role in the representation theory. However, the algorithm given by Kazhdan and Lusztig to compute these cells is enormously complicated. These cells have been worked out only in a very few cases. In the present thesis, we shall find all the left, right and two-sided cells in the affine Weyl group A n of type A n - 1 > 2. Our main results show that each left (resp. right) cell of A n determines a partition, say λ of n and, is characterized by a λ-tabloid and also by its generalized right (resp. left ) T-invariant. There exists a one-to-one correspondence between the set of two-sided cells of A n and the set A n of partitions of n. The number of left (resp. right) cells corresponding to a given partition λ ϵ A n is equal to n l / m п j = 1 u j l , where {u 1 > … > um} is the dual partition of λ. Each two- sided cell in A n is also an RL-equivalence class of A n and is a connected set. Each left (resp. right) cell in A n is a maximal left (resp. right) connected component in the two-sided cell of A n containing it. Let P be any proper standard parabolic subgroup of A n isomorphic to the symmetric group S n - then the intersection of P with each two-sided cell of A n is non-empty and is just a two-sided cell of P. The intersection of P with each left (rasp, right) cell of A n is either empty or a left (resp. right) cell of A n. Most of these results were conjectured by Lusstig [2], [3].

Item Type:	Thesis or Dissertation (PhD)
Subjects:	Q Science > QA Mathematics
Library of Congress Subject Headings (LCSH):	Weyl groups, Coxeter groups
Official Date:	1984
Dates:	Date Event 1984 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Mathematics Institute
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Carter, Roger W. (Roger William)
Sponsors:	China. Government ; Committee of Vice-Chancellors and Principals of the Universities of the United Kingdom, sponsor.
Extent:	233 leaves : charts
Language:	eng

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