Abstract
Admissible W-graphs were defined and combinatorially characterised by Stembridge in reference [12]. The theory of admissible W-graphs was motivated by the need to construct W-graphs for Kazhdan-Lusztig cells, which play an important role in the representation theory of Hecke algebras, without computing Kazhdan-Lusztig polynomials. In this paper, we shall show that type A-admissible W-cells are Kazhdan-Lusztig as conjectured by Stembridge in his original paper.
Highlights
Let (W, S) be a Coxeter system and H(W ) its Hecke algebra over Z[q, q−1], the ring of Laurent polynomials in the indeterminate q
We are interested in W-graphs corresponding to Kazhdan–Lusztig left cells
In principle, when computing left cells one encounters the problem of having to compute a large number of Kazhdan–Lusztig polynomials before any explicit description of their W-graphs can be given
Summary
Let (W, S) be a Coxeter system and H(W ) its Hecke algebra over Z[q, q−1], the ring of Laurent polynomials in the indeterminate q. In [4], Chmutov established the first step towards the proof of Stembridge’s conjecture, showing that if (W, S) is of type An−1 the simple part of a W-molecule is isomorphic to the simple part of a Kazhdan–Lusztig left cell. We are able to use the combinatorics of tableaux to show that type A admissible W-graphs are ordered, in the sense of Definition 8.1: if an arc has tail corresponding to a standard tableau t and head corresponding to a standard tableau u either u < t, or else head and tail belong to the same molecule and u = st for some simple transposition s This property of admissible W-graphs is the key to our proof of the conjecture of Stembridge.
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