TITLE: Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs)?
QUESTION [9 upvotes]: There are a couple of indirect methods, using Lie theory or Springer's general theory of regular elements in (real, complex) reflection groups, to construct natural embeddings among certain Weyl groups.    But it should be possible to find such explicit embeddings in more elementary sources.   One starts with the familiar finite real reflection groups of types $A_n \:(n \geq 2), D_n \: (n \geq 4), E_6$ and their Coxeter graphs (which are also Dynkin diagrams).    Each graph has an obvious "folding" which gives respectively reflection groups of types: $BC_\ell$ (with $\ell = n/2$ if $n$ is even or $(n+1)/2$ if $n$ is odd); $BC_{n-1}$; and $F_4$.  Here $BC_\ell$ is the Weyl group of Lie type $B_\ell$ or  $C_\ell$ and has a normal subgroup $(\mathbb{Z}/2\mathbb{Z})^\ell$ acted on by $S_\ell$.

Is there an elementary construction in the literature of such embeddings of finite reflection groups?

For example, the reflection group of type $E_6$ has order $2^7 \; 3^4 \; 5$ and is realized (in Bourbaki or the Atlas of finite Groups) in various ways, for instance as the automorphism group of the 27 lines on a cubic surface having a simple subgroup of index 2.  On the other hand, the group of type $F_4$ has order $2^7 \; 3^2$ and also has some standard realizations.

REPLY [3 votes]: One reference for these embeddings is the first section of
[Stein] Robert Steinberg, Endomorphisms of linear algebraic groups.
       Memoirs of the American Mathematical Society, No. 80
This does not quite answer Jim Humphreys' question, as Steinberg makes use of Lie theory.  I mention it mainly because of the above comments poining to:
[Stem] John Stembridge, Folding by Automorphisms.
         http://www.math.lsa.umich.edu/~jrs/papers/folding.pdf
Unlike the results in [Stem], the results in [Stein] also cover the folding $A_{2n} \leadsto  BC_n$.
Brief summary:  Consider an automorphism $\sigma$ of a Dynkin diagram $A$, the induced folding $A \leadsto A^{\text{folded}}$ and the corresponding embedding of groups $W^{\text{folded}} \hookrightarrow W$.  The nodes of $A^{\text{folded}}$ correspond to the orbits of $\sigma$ in $A$.  Denote the generator of $W$ corresponding to a node $\alpha\in A$ as $s_\alpha$, and the generator of $W^{\text{folded}}$ corresponding to a $\sigma$-orbit $B$ as $s_B$.  Claim 3 in [Stem] asserts that the embedding $W^{\text{folded}} \hookrightarrow W$ has the form
$$
s_B \mapsto \prod_{\alpha\in B}s_\alpha
$$
provided that each $\sigma$-orbit is an edge-free set in $A$.
This covers the cases $E_6\leadsto F_4$, $D_n \leadsto BC_{n-1}$ and $A_{2n-1}\leadsto BC_n$.  Proposition 1.30 in [Stein] shows more generally that this embedding has the form
$$
s_B \mapsto w_0^B,
$$
where $w_0^B$ denotes the longest word in the parabolic subgroup corresponding to $B\subset A$. Specifically, according to this result the embedding for $A_{2n}\leadsto BC_n$ has the form
$$
s_{\{i,2n-i\}} \mapsto
\begin{cases}
s_is_{2n-i}  & \text{ for } i = 1,\dots,n-1 \\
s_n s_{n+1} s_n & \text{ for } i = n-1
\end{cases}
$$