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The Kazhdan-Lusztig basis and the Temperley-Lieb quotient in type \(D\). (English) Zbl 0969.20003

The author studies an IC basis (as defined by J. Du [in CMS Conf. Proc. 14, 165-174 (1993; Zbl 0827.20052)]) for a Temperley-Lieb type quotient of the Hecke algebra of type \(D\). The main result is that this basis agrees with the projection of those Kazhdan-Lusztig basis elements \(C_w'\) indexed by elements \(w\) of the Weyl group all of whose reduced expressions avoid substrings of the form \(s_is_js_i\). The resulting basis of the quotient is cellular in the sense of J. J. Graham and G. I. Lehrer [Invent. Math. 123, No. 1, 1-34 (1996; Zbl 0853.20029)], which is implicit in the work of C. K. Fan [J. Am. Math. Soc. 10, No. 1, 139-167 (1997; Zbl 0861.20042)].
The result of this paper is a generalization to type \(D\) of a result proved by the author and the reviewer in type \(B\) [Int. Math. Res. Not. 2000, No. 1, 23-34 (2000; Zbl 0961.20008)]. However, the theorem is more difficult to prove in type \(D\) due to the less favourable combinatorial properties of reduced expressions. Another complication is that the kernel of the map from the Hecke algebra to the Temperley-Lieb type quotient is not compatible with the Kazhdan-Lusztig basis.

MSC:

20C08 Hecke algebras and their representations
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