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Left cells

Jim Humphreys asked me if I could help him visualize the left cells in the affine Weyl group A3~. This is not the place to go into details of what these objects are, but perhaps a little elaboration is useful.

The usual Weyl group A3 is a finite group with an action on a three dimensional real vector space V. In fact, the group is the symmetric group S4, and the vector space is the orthogonal complement in R^4 to the vector (1,1,1,1), where S4 acts by permuting the coordinates. There is a decomposition of V into 24 simplicial cones ("chambers"), and each chamber is further subdivided into congruent simplices ("alcoves"). One chamber is labelled the "dominant chamber." Then Lusztig defined a partition of the dominant chamber into a finite collection of sets of alcoves, and these sets are the "left cells." The actual definition of these cells is quite complicated (it involves the Kazhdan-Lusztig polynomials), so for that one must check the literature. We just mention one result about them: for the group of type An~, the left cells are in bijection with the partitions of n+1.

To compute the cells, we can use results of Shi, who developed some intricate combinatorial machinery to describe the cells in the A series. In particular, the decomposition into alcoves is induced by an affine hyperplane arrangement H in V, and one can encode which alcoves are in which cells by attaching a list of "sign conditions" to each cell. (These sign conditions simply record where an alcove sits relative to a distinguished subset of H.) Shi actually wrote the sign conditions out explicitly for A3~, so it's just a matter of figuring out what things look like.

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Revised: Fri Oct 1 22:49:08 EDT 2010
Paul Gunnells
gunnells at math dot umass dot edu