More on Coxeter Groups

Definition 1.  Let $B$ and $N$ be subgroups of a group $G.$  They are called a BN-pair if

1. $B\cup N$ generates $G$ and $B\cap N\unlhd N.$
2. $W=N/(B\cap N)$ is generated by a set $S$ of involutions.
3. Let $s\in S$ and $w\in W,$ then $BsB\cdot BwB\subseteq BswB\cup BwB.$
4. Let $s\in S,$ then $BsB\cdot BsB\neq B.$

$W$ is called the Weyl group of the BN-pair, and $|S|$ is called the rank of the BN-pair.

So from 3 and 4 we have $BsB\cdot BsB\subseteq BsB.$  If $G$ has a BN-pair, then it has a direct decomposition called the Bruhat decomposition.  It has the form

$\displaystyle G=\bigsqcup_{w\in W}BwB.$

This is plausible since $W$ has stuff in $N$ but not in $B$ (if not $1$).  So a term in the union is represented as a product of something in $N$ (but not $B$) multiplied on both sides by something in $B,$ but together $B$ and $N$ generate $G.$

One can further show that if $G$ has a BN-pair, then $W$ is a Coxeter group with generating set $S.$  Also, every element by definition has the form $g=r_1\cdots r_k$ with $r_i$ generators and $k$ minimal.  $k$ is then called the length of $g$ and denoted $\ell(g).$

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

Exchange Property.  Let $g=s_1\cdots s_k$ be a reduced word with each $s_i$ a generator.  If $\ell(sg)\leq\ell(g)$ for a generator $s,$ then $sg=s_1\cdots\hat{s_i}\cdots s_k$ for some $i\leq k.$

We could replace the generator $s$ with a reflection $t$ 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 $g=s_1\cdots s_k$ and $\ell(g) then $g=s_1\cdots\hat{s_i}\cdots\hat{s_j}\cdots s_k$ for $1\leq i

Theorem 4.  Let $(G,S)$ be a pair with $G$ a group and $S$ a generating set of $G$ such that $s^2=1$ for all $s\in S.$  Then the following are equivalent.

1. $G$ is a Coxeter group.
2. $(G,S)$ satisfies the exchange property.
3. $(G,S)$ satisfies the deletion property.

Also worthy of mentioning is the concept of descents.  Let $(G,S)$ be a pair as described above and $T$ be the set of reflections in $G.$

Definition 5.  Define

$T_L(g)=\{t\in T:\ell(tg)<\ell(g)\}$

$T_R(g)=\{t\in T:\ell(gt)<\ell(g)\}$

and then define $D_L(g)=T_L(g)\cap S$ and $D_R(g)=T_R(g)\cap S.$  The latter two sets are respectively called the left and right descents of $g.$

My initial terminological intuition is that $s\in D_L(g)$ is a left descent in the sense that the “altitude” relative to $g$ is lower when shifted to $s$: $\ell(sg)<\ell(g).$

Now let $g=s_1\cdots s_k$ be reduced.  Then $\ell(g)=k.$  Moreover left multiples of $g$ that shorten its length must have the form $s_1s_2\cdots s_i\cdots s_2s_1$ where $1\leq i\leq k.$  So $T_L(g)$ are precisely those elements.  There are $k$ such palindrome killers.  So we have $|T_L(g)|=\ell(g).$The palindrome notion is clear.  The killer part refers to the palindrome killing more letters than it adds–namely killing $i$ letters while adding $i-1$ letters (hence net kill $1$).

Since the inverse of an element is its mirror, it follows that $T_L(g)=T_R(g^{-1})$ and $\ell(g)=\ell(g^{-1}).$  We also thus have $|T_L(g^{-1})|=\ell(g).$  This is easy to see as palindrome killers of the inverse have the form $s_ks_{k-1}\cdots s_{k-i}\cdots s_{k-1}s_k$ with $0\leq i\leq k-1.$

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