Open AccessDissertation
The decomposition into cells of the affine Weyl groups of type A
TLDR
In this article, it was shown that the left, right and two-sided cells in the affine Weyl group A n of type A n - 1 > 2 are T-invariant.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].read more
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Journal ArticleDOI
Left cells in affine Weyl groups
TL;DR: In this article, the authors prove a property of left cells in affine Weyl groups W, by which they formulate an algorithm to find a representative set of the left cells of W in any given two-sided cell.
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Cells in Coxeter groups I
TL;DR: In this article, the combinatorial structure of Kazhdan-Lusztig cells in infinite Coxeter groups was studied. But the main focus was on the set of distinguished involutions in the Coxeter group.
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On the determination of Kazhdan-Lusztig cells in affine Weyl groups with unequal parameters
TL;DR: In this article, it was shown that there are only finitely many possible decompositions into left cells and the number of left cells is finite in each case, thus confirming some of Lusztig's conjectures.