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