Reference for embeddings of reflection groups (related to folding ADE Coxeter graphs)?

Question

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.

Answer 1

Rather than overload the question with side remarks, I'll add some background here in community-wiki format to indicate what can be gotten from Springer's relatively elementary treatment of regular elements here. ("Regular elements of finite reflection groups", Invent. Math. 25 (1974), 159-198.)

He deals in great generality with finite complex (= unitary) reflection groups, including all finite Coxeter groups (= real reflection groups), the Weyl groups being the most important of these. Besides the structure and classification results of Coxeter and Shephard-Todd, he relies mainly on Chevalley's treatment of invariants in the standard matrix realization.

Folding of Coxeter diagrams comes up just in the case of the Weyl groups $W$ which I listed (though less frequently for $A_n$ when $n$ is even). Lie theory and the crystallographic root systems aren't really needed for my question, except perhaps for type $D_n$ with $n$ even. Consider the very special case of regular elements $w$ of order $d=2$ in Springer's paper, which exist and are all conjugate. Write the list of degrees of fundamental invariants as $d_1, \dots, d_n$. Of these the even ones (those divisible by $d$) are the degrees of the centralizer of $w$ in $W$, itself a finite (real or complex) reflection group.

Leaving aside the case of $D_n$ for even $n$, my list of types matches those $W$ not containing $-1$ as longest element $w_\circ$. On the other hand, it's easy to see that $w_\circ$ is always regular (for any finite Coxeter group). Since $-1 \in W$ iff all $d_i$ are even, the centralizer of our regular $w_\circ$ is a proper subgroup having the degrees of the Coxeter group obtained by folding and is in fact that subgroup (easy to check how the eigenvalues $\pm 1$ behave).

As in my question, the isolated case $E_6 \rightsquigarrow F_4$ gives a nice example of the resulting subgroup embedding. Have these embeddings been written down explicitly?

Answer 2

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 $$ s_{\{i,2n-i\}} \mapsto \begin{cases} s_is_{2n-i} & \text{ for } i = 1,\dots,n-1 \\ s_n s_{n+1} s_n & \text{ for } i = n-1 \end{cases} $$