María de la Paz Quirós Artacho: Coxeter groups, Hecke algebras and Kazhdan-Lusztig cells

Abstract

Coxeter groups arise in many different areas of mathematics and are extensively studied in algebra, geometry, and combinatorics. One of their important areas of applications is by means of their associated Hecke algebras, which are certain deformations of the group algebras of Coxeter groups that play an important role in representation theory. A turning point in the study of representations of Hecke algebras was the celebrated paper Representations of Coxeter groups and Hecke algebras by Kazhdan and Lusztig where the notions of left, right and two-sided cells of an arbitrary Coxeter group, now called Kazhdan-Lusztig cells, were first introduced. Their definition incorporates a new canonical basis of the Hecke algebra, the Kazhdan-Lusztig basis, and they give rise to representations of both the Coxeter group itself and the associated Hecke algebra.

In this thesis we start with an introduction of Coxeter groups, focusing on some structural aspects of its rich theory that are of combinatorial, algebraic and geometric interest. We then move on to study their associated Hecke algebras, starting from a more general construction of associative algebras over a commutative ring, leading towards the construction of the Kazhdan-Lusztig basis and the study of the action of the canonical basis of the Hecke algebra on the Kazhdan-Lusztig basis, which turns out to be key in the determination of the partition of the Coxeter group into Kazhdan-Lusztig cells as well as the properties of the partition. We then focus on the study of Kazhdan-Lusztig cells and discuss several tools that allow us to deduce deep properties about Kazhdan-Lusztig cells as well as a series of related conjectures by Lusztig.

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