Coxeter Groups(intro)

Reflection groups

In order to understand the intuition underlying the theory of Coxeter groups(and Weyl groups in particular groups in particular), you can go through the theory of reflection groups which I’ll only superficially treat preceding my exposition of Coxetr systems and the associated representation theory.

Consider some Euclidean space $V$ and a vector $\alpha \in V$ . A reflection associated with $\alpha$ is a linear operator which sends $\alpha$ to $-\alpha$ and point-wise fixes the orthogonal hyperplane/subspace $H_{\alpha}$ .

If the reflection is $s_{\alpha}$ , then it can represented as:

$s_{\alpha}(x)=x-\frac{2(x,\alpha)}{(\alpha,\alpha)}\alpha=x-2 proj_{\alpha}(x)$

Clearly $s_{\alpha}$ is an orthogonal transformation and the set of all reflections in $V$ can be recognized as the subgroup of $O(V)$ (orthogonal group of $V$ ) consisting of elements of order 2.

I recommend checking out Humphrey’s book on Reflection groups to get a basic idea on root systems and simple systems which will be useful in the discussion about Weyl groups.

One example is the group $S_{n}$ which is generated by the set of transpositions $(i,i+1)$ for all $1 \leq i \leq n-1$ (a simple exercise), hence making it a reflection group. Let $s_{i}=(i,i+1)$ . Here, the elements satisfy the following relations:

$s_{i}^{2}=1$
$s_{i}s_{j}=s_{j}s_{i}$
$s_{i}s_{i+1}s_{i}=s_{i+1}s_{i}s_{i+1}$

The last relation is called a braid relation and is a key concept in the definition of Coxeter groups. These relations, when generalized in a particular manner, yield Coxeter groups.

Consider some simple system $\Delta$ and let $W$ be the group generated by the simple reflections. For some $w \in W$ , let $w=s_{\alpha_{1}}....s_{\alpha_{k}}$ where $\alpha_{i} \in \Delta$ and repetitions are allowed.

The length of $w,l(w)$ is defined as the as the smallest value of $k$ for which such an expression exists.

Coxeter Groups

A Coxeter group $G$ is a group with a set of generators, $S=\{s_{1},s_{2},\cdots,s_{n}$ satisfying:

$(s_{i}s_{j})^{m(i,j)}=1$ where $m(i,j)=1$ if $i=j$ and $m(i,j) \geq 2$ if $i \neq j$ . If no relation exists between $s_{i}s_{j}$ ,then $m(i,j)$ is represented as $\infty$ .

here, $m(i,j)$ is the order of $s_{i}s_{j}$ . if $m(i,j)=2$ ,then $s_{i}s_{j}=s_{j}s_{i}$ since all $s \in S$ have order 2. For order 3, the relation is $s_{i}s_{j}s_{i}=s_{j}s_{i}s_{j}$ . These are simply the braid relations which we discussed before.

The pair $(W,S)$ is called a Coxeter system. The matrix $M=(m(i,j))$ associated by the natural indexing through the set $S$ is called the Coxeter matrix and is obviously symmetric.

Example: Consider the set of generators $S=\{s_{1},s_{2},s_{3} \}$ with $m(s_{1},s_{2})=3,m(s_{1},s_{3})=2,m(s_{2},s_{3})=\infty$ . let $W$ be the generated Coxeter group.

Map the generators $s_{1},s_{2},s_{3}$ respectively to the following elements in $PGL(2;\mathbb{Z})$ .

$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} -1 & 1 \\ 1 & 0 \end{bmatrix} \begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}$

By calculating the order of the products of these generators, we see that there is a homomorphism $\psi:W \mapsto PGL(2,\mathbb{Z})$ .

Note that $SL(2;Z)$ is generated by the two elements of the form:

$\begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \begin{bmatrix} 1 & 1 \\ 0 & 1 \\ \end{bmatrix}$

Hence the image set $\psi(W)$ includes PSL(2,\mathbb{Z}) as a subgroup. Also, $PSL(2,\mathbb{Z})$ has index 2.Note that the elements $s_{1},s_{2},s_{3}$ all have determinant $-1$ . So, the images of these elements under the homomorphism isn’t included in $PSL(2,\mathbb{Z})$ . Hence, the homomorphism is surjective. It is a simple exercise to prove that $ker(\psi)=\{\phi \}$ . Hence, $W \cong PGL(2,\mathbb{Z})$ .

Lemma:

There is a homomorphism $\epsilon:W \mapsto \{-1,+1\}$ given by $\epsilon(w)=(-1)^{l(w)}$

Proof:

It is clear that $l(w)=l(w^{-1})$ for all $w \in Q$ . For $w,w' \in W$ , their reduced expressions can be represented as $w=s_{1}....s_{k_{1}}$ and $w'=s'_{1}....s'_{k_{2}}$ . Hence the maximum value that can be attained by $l(ww')=k_{1}+k_{2}$ . Hence, $l(ww') \leq l(w)+l(w')$ .

Taking $l(ww'w'^{-1}) \geq l(ww')+l(w'^{-1}) \Rightarrow l(ww') \geq l(w)-l(w')$ .

So replacing $w'=s$ for some $s \in S$ , we get $l(w)-1 \leq l(ws) \geq l(w) +1$ . (1)

Now, consider some $w\in W$ and a reduced expression $w=s_{1}....s_{k}$ .

$\epsilon(w)=\epsilon(s_{1}) \cdots \epsilon(s_{k})=(-1)^{k}=(-1)^{l(w)}$ . So, $\epsilon(ws)=-\epsilon(w)$ . From equation (1), $l(ws)=l(w) \pm 1$ .

Now since we have loosened the definitions and generalized reflection groups to Coxeter groups, we will need to redefine the machinery underlying reflections of a vector space.

Assume that $V$ is a vector space over $\mathbb{R}$ having a basis $\beta=\{\alpha_{s}|s \in S \}$ So, the elements are in one-to-one correspondence with the generators. Since we are dealing with a discrete indexed set and not a continuum of vectors in a Euclidean space, we will need a better way to understand ‘angles’ between $\alpha_{s}$ and $\alpha_{s}'$ . So we define a bilinear form $B$ on the vector space $V$ as follows:

$B(\alpha_{s},\alpha_{s}')=-cos\frac{\pi}{m(s,s')}$ .

This seems like a legitimate definition. Connecting the order of $ss'$ to the represent an ‘angle’.

Note that $B$ is interpreted to be -1 if there is no relation between $s,s'$ i.e $m(s,s')=\infty$ .

Clearly, $B$ is also symmetric. With this symmetric bilinear form, we can define reflections $\sigma$ associated with the generators as follows

$\sigma_{s}(x)=x-2B(x,\alpha_{s})\alpha_{s}$ . Again, similar to the reflections described earlier, $\sigma_{s}$ point-wise fixes the hyperplane $H_{s}$ .

Proposition I:

$\sigma_{s}$ preserves the bilinear form $B$ on $V$ i.e $B(\sigma_{s}a,\sigma_{s}b)=B(a,b) \forall a,b \in V$ . Hence, the subgroup generated by all reflections $\sigma_{s},s \in S$ preserve $B$ .

Let $V_{s,s'}=\mathbb{R}\alpha_{s} \oplus \mathbb{R}\alpha_{s'}$ , then the reflections $\sigma_{s},\sigma_{s'}$ clearly stabilize $V_{s,s'}$ , as can be verified through simple calculations.

Also, restricting the symmetric bilinear form $B$ to $V_{s,s'}$ makes it positive-semidefinite. To prove this, just take an arbirtary element element $k \in V_{s,s'}$ given by $k=a\alpha_{s}+b\alpha_{s'}$ where $a,b \in \mathbb{R}$ and expand $B(k,k)=B(a\alpha_{s}+b\alpha_{s'},a\alpha_{s}+b\alpha_{s'})$ bilinearly to obtain

$B(k,k)=a^{2}+b^{2}-2ab.cos(\frac{\pi}{m(s,s')}) =(a-bcos(\frac{\pi}{m(s,s')}))^{2}+b^{2}sin^{2}(\frac{\pi}{m(s,s')}) \geq 0$ .

All this this culminates into a very basic theorem, yielding the representation of the Coxeter group.

Theorem:

There is a unique homomorphism $\sigma:W \mapsto GL(V)$ which maps $s$ to $\sigma_{s}$ and the image, $\sigma(W)$ preserve the symmetric bilinear form $B$ on $V$ .

This group representation is commonly also referred to as the geometric representation of $W$ , the Coxeter group.

This is a very basic and cursory introduction to reflection groups and Coxeter groups which is all I had time to flesh out this week due to other obligations. In following posts, I plan to expand on various topics including Coxeter-Dynkin diagrams, Kazdhan-Lusztig polynomials and especially the extremely important study of affine Weyl groups and Iwahori-Hecke algebras which I’ve felt is a grossly criminal offense to omit from my exposition on Coxeter groups. I’ve also been reading a few papers on the rich theory of modular representations of Hecke algebras and modern results in connection to cellular algebra and the study of combinatorics of tableaux which I’ll promptly delineate along with a bit of discussion on state-of-the-art results in the field.

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Reflection groups

Coxeter Groups

Lemma:

Proposition I:

Theorem:

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