1

Let $E$ be the subspace of $\mathbb{R}^{n+1}$ for which the coordinates sum to $0$ and let $\Phi$ be the set of vectors in $E$ of length $\sqrt{2}$ and which are integers vectors.

It is known that $\Phi$ is a root system of finite type $A_n$. One choice of simple roots expressed in the standard basis is: $\alpha_i=e_i-e_{i+1}$, for $1\leq i \leq n$. The reflection w.r.t. $\alpha_i$ permutes the subscripts $i, i+1$ and leaves all other subscripts fixed. Thus $\sigma_{\alpha_i}$ corresponds to the transposition $(i,i+1)$ in the symmetric group $S_n$. These transpositions generate $S_{n+1}$, so we obtain a natural isomorphism of the Weyl group $W_{\Phi}$ corresponding to the root system $\Phi$ onto $S_{n+1}$.

The first question is the following:

In order to compute all the elements of $W_{\Phi}$, can I just express each element of $S_{n+1}$ in terms of the adjacent transpositions and then use the bijection to map it to the element (reduced word in simple reflections) in $W_{\Phi}$?

Also, in wiki entry on Weyl groups I read that the Weyl group for $A_n$ is the permutation group on $n+1$.

Probably a silly question, but why don't they just say symmetric group instead of permutation group? What is the distinction here?

Shaun
  • 44,997
billy192
  • 405

1 Answers1

2

First question: Yes.

Second question: I've heard both names used. I guess by "permutation group" people might more generally mean a subgroup of some $S_m$ in some contexts, but in this case, it's definitely the "full" permutation group / symmetric group $S_{n+1}$. (By the way I think your term "symmetry group" is also non-standard, as it refers to the group of all symmetries of a geometric object. Incidentally, the symmetry group of the root system $A_{n}$ is larger than its Weyl group for $ n\ge 2$.)

Torsten Schoeneberg
  • 26,051
  • Thanks! Yeah, that's what was confusing me: I think of permutation group as some subgroup of some symmetric group, but I guess subgroup can be the whole thing, which is the case here. As for the additional comment you made: for $A_2$ root system its Weyl group, geometrically, is the symmetry group of an equilateral triangle. Does the $A_n$ Weyl group has this sort of geometric realization for all $n$? – billy192 Mar 23 '20 at 15:38
  • 1
    I'm sure there's geometric objects which have $S_n$ as symmetry group (like, $n$ rays pointing from the origin at same angles), and you coould ask that as a new question, but I think that i sleading us far from the original question, and could be really counterproductive in understanding root systems and their symmetries and Weyl groups. – Torsten Schoeneberg Mar 24 '20 at 03:48
  • 1
    Update on your question in the comment: Look at https://math.stackexchange.com/q/46317/96384, in particular the answer by Vladimir Sotirov, for realisations of symmetric groups as symmetry groups. For more in that vein, cf. https://math.stackexchange.com/q/1252746/96384 and https://mathoverflow.net/q/993/27465. But, as said, this leads us quite far away from Weyl groups and root systems. – Torsten Schoeneberg Mar 25 '20 at 04:47
  • Thanks! That is very useful! – billy192 Mar 25 '20 at 16:42