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Results tagged with gr.group-theory
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user 21291
Questions about the branch of algebra that deals with groups.
6
votes
Accepted
Elements of Coxeter group whose simple reflections pairwise commute
If I see it correctly, there is not much going on in the set $X(G)$ from the viewpoint of Coxeter systems: Let $S$ be the simple system and let $G$ be the Coxeter graph of $(W,S)$ with vertex set $S$. …
- 3,271
3
votes
Accepted
Do we have a one to one correspondence between positive roots and reflections in a Coxeter g...
I suggest, you really look carefully into the reference "Humphreys, Reflection and Coxeter groups" (link behind paywall), as you again find the answer there.
Please look into the reference for details …
- 3,271
7
votes
Accepted
References request: reflections in coxeter groups
As you find in "Humphreys, Reflection and Coxeter groups" (link behind paywall) in Section 5.7, the set of reflections of a Coxeter system $(W,S)$ is given by $R = \{ wsw^{-1} : w \in W, s \in S\}$, t …
- 3,271
6
votes
A decomposition of $w_0$ which is similar to the reduced decomposition
Consider the (strong) Bruhat order on the symmetric group: This order is defined for $a,b \in \mathcal{S}_n$ as $a \leq b$ if $\ell(a) < \ell(b)$ and $at=b$ for a transposition $t$.
Therefore, your f …
- 3,271
4
votes
Uniform proof that a finite (irreducible real) reflection group is determined by its degrees?
As it is not clearly stated here so far: As far as I know there is still no conceptual explanation of this observation.
We were looking at this situation for unitary reflection groups in Proposition 2 …
- 3,271