# 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.

# The Relentless Ingroup Bias

With the termination of the military’s “Don’t Ask Don’t Tell” policy as of yesterday, I am reminded of the fundamental concept at play that delayed this resolution for so long. This fundamental concept is also at the core of many other issues: gender discrimination, racial discrimination, religious discrimination,…, X discrimination. This fundamental concept is at the heart of contemporary political partisan bickering, the wealth gap, and wars altogether. This fundamental concept is the scarcity of resources. When the set of resources available to a population in insufficient for that population, members of the population will inevitably compete for them. The ingroup bias has become the catalyst for nonuniform distribution of resources.

The ingroup bias follows from the ingroup instigation. The heuristic for obtaining resources is cooperative gameplay. We primitively wish to form alliances with people for the sole purpose of overpowering others (or groups of others) in order to ensure the acquisition of resources. This is the biologically intrinsic (and evolutionally reinforced) tendency which I call the ingroup instigation. Given the objective of the ingroup (to work together), the ingroup bias (the tendency to favor individuals in the ingroup and disfavor those in the outgroup) follows. If we assume this to be true, then the function determining which individuals will group is only governed by what best allows them to overpower other groups or individuals. By default this starts with proximity and special cases of it such as family (note also how fundamental physical forces operate). Then, once some groups gather many resources, it may be beneficial for them to team as well. This can be seen in political alliances, corporate mergers, residential segregation with respect to socioeconomic status (i.e. poor towns or rich towns),…,collisions of galaxies, etc.

It then follows that any minority or group of individuals who have less power become targets of the majority, simply because it is easy to take their resources. Whether it be a minority based on gender, race, sexual orientation, or religion, the characteristic of a group being a minority–not being in the ingroup of those with the quantitative or qualitative power–becomes sufficient for hindering their progress and in turn targeting their resources.

Hopefully one day we will realize that “potential knowledge” is a good with no scarcity that can in turn be distributed to everyone without limits. The amount of knowledge in a system will always be finite (but not constant), yet it will also always be an upper bound on usable resources in society (i.e. the amount of consumable resources in the system is dependent upon the amount of knowledge [on how to create such resources from raw resources] in the system at that time).

# 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.