Abstract
The Hecke algebra associated to a Coxeter system (W, S) is a \(\mathbb {Z}[v,v^{-1}]\)-algebra that recovers the group algebra \(\mathbb {Z}[W]\) under the specialization v↦1. It has two important bases: the standard basis, and a certain “self-dual” basis called the Kazhdan–Lusztig basis, with the change of basis between the two given by the Kazhdan–Lusztig polynomials. In this chapter, we briefly review the definition and construction of the Hecke algebra and its Kazhdan–Lusztig basis.
This chapter is based on expanded notes of a lecture given by the authors and taken by
Joel Gibson and Alexander Kerschl
J. Gibson ⋅ A. Kerschl
School of Mathematics and Statistics, University of Sydney, Sydney, NSW, Australia
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Notes
- 1.
We leave the reader to confirm that this argument still works when w = x, despite the superficial difference in (3.26).
- 2.
Or should that be “mu”?
- 3.
In [59], a stroll is called a Bruhat stroll.
- 4.
Note that \(\ell ({ \underline {x}})\) is the length m of the sequence \({ \underline {x}}\), not the length of the element x it expresses.
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Elias, B., Makisumi, S., Thiel, U., Williamson, G. (2020). The Hecke Algebra and Kazhdan–Lusztig Polynomials. In: Introduction to Soergel Bimodules. RSME Springer Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-48826-0_3
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DOI: https://doi.org/10.1007/978-3-030-48826-0_3
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