hecke algebra – Weekly Mathematics

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.

Continue reading “Coxeter Groups(intro)”