# Hecke Algebra of a Coxeter Group

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.

# Bruhat Order

The **Bruhat order** is defined on a Coxeter group in the following manner for :

- with a reflection (conjugate of a generator) and ,

Reflexivity is clear since (assuming is considered a generator). Transitivity is trivial. And if we have and then in the simple case where and we have

and

which imply that and hence that So Since we must have it follows that The general cases are done by induction. So we have antisymmetry.

So it is a partial order on the Coxeter group. The **Bruhat digraph** is constructed with elements as vertices and a directed edge from to iff Note how this ordering differs from the simplicial ordering we previously mentioned on the induced Coxeter complex.

Recall the notion of standard subgroups (also known as parabolic subgroups) of a Coxeter group where triply represents a subset of indices of a generating set, the corresponding generators indexed by those indices, and the subgroup generated by such generators. Also recall the definition of descents of elements. In the following definition, are used in the second sense (as subsets of generators), and we will use to represent the subgroup generated by

**Definition 1.** Let and define

The sets are called the **right descent classes** (or the analogous definition for left descent classes).

**Proposition 2.** Let Then every has a unique factorization of the form where and Moreover we have

We can define the map where This map is clearly idempotent. Moreover we have the following.

**Proposition 3.** The map preserves Bruhat order

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

# More on Coxeter Groups

**Definition 1.** Let and be subgroups of a group They are called a **BN-pair** if

- generates and
- is generated by a set of involutions.
- Let and then
- Let then

is called the **Weyl group** of the BN-pair, and is called the **rank** of the BN-pair.

So from 3 and 4 we have If has a BN-pair, then it has a direct decomposition called the **Bruhat decomposition**. It has the form

This is plausible since has stuff in but not in (if not ). So a term in the union is represented as a product of something in (but not ) multiplied on both sides by something in but together and generate

One can further show that if has a BN-pair, then is a Coxeter group with generating set Also, every element by definition has the form with generators and minimal. is then called the **length** of and denoted

**Definition 2.** We define a **reflection** in a Coxeter group as a conjugate of a generator.

**Exchange Property.** Let be a reduced word with each a generator. If for a generator then for some

We could replace the generator with a reflection and remove the reduced and possible equality to obtain the **Strong Exchange Property**.

**Proposition 3.** The strong exchange property holds in a Coxeter group.

**Deletion Property.** Let and then for

**Theorem 4**. Let be a pair with a group and a generating set of such that for all Then the following are equivalent.

- is a Coxeter group.
- satisfies the exchange property.
- satisfies the deletion property.

Also worthy of mentioning is the concept of *descents*. Let be a pair as described above and be the set of reflections in

**Definition 5.** Define

and then define and The latter two sets are respectively called the **left** and **right descents of**

My initial terminological intuition is that is a left descent in the sense that the “altitude” relative to is lower when shifted to :

Now let be reduced. Then Moreover left multiples of that shorten its length must have the form where So are precisely those elements. There are such* palindrome killers*. So we have The palindrome notion is clear. The killer part refers to the palindrome killing more letters than it adds–namely killing letters while adding letters (hence net kill ).

Since the inverse of an element is its mirror, it follows that and We also thus have This is easy to see as palindrome killers of the inverse have the form with

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

# Metric Construction of Buildings

Let be a Coxeter group where is generated by elements. Also by for we mean the minimal length is a product decomposition of Let us first use the book definition.

**Definition 1.** A **Weyl distance function** is a map where a set whose elements are called chambers such that

- if and such that then If we also have that then
- if then for any we have a chamber such that and

The pair is called a **– metric space**. The triple is called a **building**.

I’m fairly certain I copied correctly (triple checked), but it looks like or the first sentence in at least. Of importance is the fact we previously mentioned: that chambers of a Coxeter complex coincide with elements of the Coxeter group (since they are (standard) cosets of the trivial standard subgroup). If we thus let then can be thought of as a product on By condition the element is thus not an identity with respect to this product. Also in this regard one can show that the chambers of a Coxeter complex form a building where

with being the gallery metric we previously defined.

Conversely we can say two elements in are **-adjacen**t if and **-equivalent** if they are either -adjacent or equal. If and are -equivalent, we write This is an equivalence relation since

**Proposition 2.** if one takes a generator value.

*Proof.* Suppose By part of the definition there is a such that and Thus by we have and thus

The equivalence classes under are called **-panels**. A **panel** is an -panel for some Galleries can be defined similarly with this terminology. Thus

[1] Abramenko, Peter and Kenneth Brown. *Buildings*. Graduate Texts in Mathematics. Vol. 248. Springer Science and Business Media. 2008.

# Coxeter Complexes

Let be a Coxeter group with generators and be a subset of We define the subgroup as the **standard subgroup** (i.e. hereafter we abuse notation by using interchangeably for the subset of indicies as well as the subgroup generated by the corresponding elements of ). Its cosets will be called **standard cosets of **.

**Definition 1**. Let be (standard) cosets of standard subgroups and We define the partial ordering If denotes the set of standard cosets in then we call the **Coxeter complex of** and will denote it for short.

It’s clear that is a poset since the ordering is merely reverse containment on all cosets. Elements of will be called **simplices**, maximal elements will be called **chambers**. Since is a standard subgroup with it follows that the chambers simply coincide with elements of Also if is generated by one generator, then it has two elements: Cosets of such standard subgroups have the form and are called **panels**. In the case where we call the **fundamental panel** of Also if is a chamber and is generated by a singleton, then is a **face** of Note that is the trivial minimal element, but suppose is generated by generators of , then a coset of is called a **vertex.**

Every panel is the face of exactly two chambers: panels have the form and are thus faces of and

Recall every element in a Coxeter group has the form

where for some . If is a chamber and we define the chamber then and have a panel in common: A **gallery** is a sequence of chambers such that and have a panel in common. We can thus define a metric on

This metric can be extended to any simplices, where it is the minimized version of the above taken over all chambers containing those simplices.

**Definition 2**. A **type function** is a map where denotes the vertices in such that it is a bijection on for all where are the vertices of the chamber The value (or singleton set) is called the **type** of we may dually call the **cotype** of

The standard type function on a Coxeter complex is defined chamber-wise by

by which we mean the one element of as ranges through (in the sense of which generator it excludes). Remember we use both to represent a subset of and to represent the Coxeter subgroup (aka standard subgroup) generated by the elements

We can generalize the type function on the Coxeter complex from vertices to all simplices. We simply map the simplex to the subset of to which all of its vertices are sent. Hence chambers get sent to the whole set, and have empty cotype.

**Definition 3**. Let be a simplex. We define its **link**, denoted as the set of all simplices such that and have a lower bound.

The link is clearly a subcomplex since if then for all subsets Thus the facet ordering is still transitive.

**Proposition 4.** Let be a simplex. Then (as posets).

[1] Abramenko, Peter and Kenneth Brown. *Buildings*. Graduate Texts in Mathematics. Vol. 248. Springer Science and Business Media. 2008.