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)<k, then g=s_1\cdots\hat{s_i}\cdots\hat{s_j}\cdots s_k for 1\leq i<j\leq k.

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.


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