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

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.

-
John Stembridge has written a few pages on such an elementary construction here: math.lsa.umich.edu/~jrs/papers/folding.ps.gz I don't know if that counts as "in the literature". –  Johannes Hahn Jul 29 '13 at 15:03
Foldings of geometries (including Coxeter geometries, i.e. these of finite reflection groups) are discussed in e.g. A.Pasini's book "Diagram geometries". –  Dima Pasechnik Jul 29 '13 at 15:47
A typo: the Weyl group of types B and C has $S_{\ell}$ acting on a normal subgroup $(\mathbf{Z}/(2))^{\ell}$ in the natural manner, not the other way around (as is written above, and doesn't make sense for $\ell > 2$). –  user36938 Jul 29 '13 at 17:05
@Johannes: Thanks for the reference to Stembridge's notes, which I had overlooked (though he avoids the case of type $A_n$ with $n$ even while making use of Lie theory). It's hard to define what's "in the literature", but the many lecture notes and such posted on home pages tend to be temporary and often unfindable in database searches. –  Jim Humphreys Jul 29 '13 at 20:10
I found Lusztig's construction of embeddings using the $W$-graph of a left cell containing a simple reflection enlightening. See the end of "some examples of square integrable reps of p-adic groups". For example one gets natural embeddings H3 in D6 and H4 in E8 as well as more well-known examples. –  Geordie Williamson Jul 30 '13 at 21:38

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?