Reflection in terms of simple reflections

Question

Suppose $\beta=\sum_{i=1}^ka_i\alpha_i$, where $\alpha_i$ are simple roots. Is there any easier way to write the reflection corresponding to $\beta$ say $s_{\beta}$ in terms of $s_{\alpha_i}$'s. I mean is there any formula to express $s_{\beta}$ as a product of $s_{\alpha_i}$'s.

Answer 1

I suspect your best hope is $$s_{\beta} = w s_{\alpha} w^{-1},$$ where $\beta = w(\alpha)$ for some simple root $\alpha$. It is always possible to write $\beta = w(\alpha)$ when $\beta$ is a root.

There are many possibilities for $w$ and $\alpha$. Define the depth of a positive root $\beta$ to be the smallest $k$ such that $w(\beta)$ is negative and $\ell(w) = k$. (See chapter 4 of ``Combinatorics of Coxeter Groups'' for further details.)

It's clear that simple roots have depth 1. It's not hard to show that the depth of $s_{\alpha_i}(\beta)$ is smaller than the depth of $\beta$ if $\langle \beta, \alpha_i \rangle > 0$. This provides a brute force procedure for finding $w$ (and $\alpha$) and expressing it as a product of simple reflections.

This is just an elaboration of Matt Pressland's comment. A good general formula appears to be too much to ask for. However, in type A (with the simple roots ordered in the usual way), there is a nice answer: $$\beta = \alpha_i + \cdots + \alpha_j \implies s_\beta = s_{\alpha_i} s_{\alpha_{i+1}} \cdots s_{\alpha_{j-1}}s_{\alpha_j}s_{\alpha_{j-1}}\cdots s_{\alpha_i}$$