Let be a Coxeter group with a generating set

**Definition 1.** The **Hecke algebra of ** denoted is generated by elements over the ring such that multiplication satisfies

where the order is Bruhat. It is trivial to verify that the Hecke algebra is unital with unit

**Proposition 2.** If then is invertible with inverse

*Proof.*

It follows that all elements of the form with are invertible, so the Hecke algebra is a division algebra. Moreover it is a *-algebra with involution defined by

It turns out that there is a basis of where

with and which can be proven to uniquely exist for elements which are called the **Kazhdan-Lusztig polynomials**.

[1] Björner, Anders and Francesco Brenti. *Combinatorics of Coxeter Groups*. Vol. 231. Graduate Texts in Mathematics. Springer. 2005.

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