Embed one Coxeter System into another

Question

What is a good reference that explains all the braid relations and diagrams for Coxeter systems concisely?

In particular, how do I embed $H_3$ inside $D_6$, or $H_4$ inside $E_8$? Any hints?

Answer 1

It's not clear exactly what your first sentence is asking for (there are a variety of surveys and books). For instance, are you only interested in finite Coxeter groups?

The embeddings of the noncrystallographic Coxeter groups $H_3, H_4$ indicated in the follow-up question probably originate in older work of Coxeter and du Val (though I don't recall the exact source). These embeddings have been studied over the years in different styles, but a single comprehensive reference may not exist. One direction is indicated in a series of papers by Matthew Dyer, e.g., Embeddings of root systems. I. Root systems over commutative rings, J. Algebra 321 (2009), no. 11, 3226–3248.

Another direction of research connects the embeddings with mathematical physics, sometimes in rather a computational style, e.g., Mehmet Koca, Ramazan Koç, Muataz Al-Barwani, Noncrystallographic Coxeter group $H_4$ in $E_8$,
J. Phys. A 34 (2001), no. 50, 11201–11213.

ADDED: Older references for the embeddings were given in the notes to Section 2.13 in my 1990 book Reflection Groups and Coxeter Groups, though in the published version I wrote $D_3$ where $D_6$ is intended (this and other corrections have been maintained on my homepage).

NOTE ON HISTORY: Having looked further into the tangled literature (which reinforces my decision not to become a historian of contemporary mathematics), I would replace the "probably" in my second paragraph by "possibly" given the lack of documentation.

(1) The earliest relevant paper seems to be the short note (in an obscure Japanese journal): Jiro Sekiguchi, Tamaki Yano, A note on the Coxeter group of type $H_3$., Sci. Rep. Saitama Univ. Ser. A 9 (1979), no. 2, 33-44. This follows their note on $F_4$ and both are motivated especially by invariant theory aspects. At this remove at least one author recalls they also discussed $H_4$ at that time but didn't write down the details. (2) In the short remark 3.9(b) of his 1983 paper linked above, Lusztig applied some of the previous Hecke algebra and $W$-graph development to the embedding of $H_4$ into $E_8$. His proof comebines elementary and more sophisticated arguments. The starting point is an assignment of vertices in the $H_4$ diagram to nonconnected pairs of vertices in the $E_8$ diagram, pictured via a "folding" of the latter graph; this induces a homomorphism of one Coxeter group into the other. (3) A unified treatment of embeddings of non-crystallographic finite Coxeter groups is given more expansively in $\S2$ of a Russian paper, with English translation: O.P. Shcherbak, Wavefronts and reflection groups, Russian Math. Surveys 43:3 (1988), 149-194. See Theorem 2 and the ``folding'' diagrams there for $H_2$ (dihedral of order 10), $H_3, H_4$ as well as comments on embedding other non-crystallographic dihedral groups. His method is mostly different from Lusztig's, whose work he may or may not have been aware of (but doesn't cite); he does acknowledge the note by Sekiguchi-Yano. Shcherbak's work is in the mode of the Arnol'd-Brieskorn work on singularities and extends it to non-crystallographic cases.

While I referred to (1) and (3) in my 1990 book, I overlooked Lusztig's remark at the time even though his paper is in the reference list. (Some of Scherbak's earlier work on $H_4$ appeared only in a non-translated short note in the same Russian journal in 1984. It is this work which figures in later developments by B. Dubrovin and others on classification of 2D topological field theories, etc.)

Answer 2

(This is more of a longer comment without a proper answer to the question.)

I assume you mean "An embedding $(W,S) \hookrightarrow (W',S')$ is an injective map from $S$ to the set $T' = \{ w s w^{-1} : w \in W', s \in S' \}$ of reflections in $W'$ generating a Coxeter system isomorphic to $(W,S)$".

• Is that what you mean?

If yes, this is the same as asking which reflection subgroups are there in $(W',S')$ that are isomorphic to $(W,S)$. By "Dyer - Reflection subgroups of Coxeter systems" (and in parallel by a paper of Dheodar), every reflection subgroup it itself a Coxeter group with canonical Coxeter generators depending on $S'$. This gives a (computationally feasible) way to systematically search for reflection subgroups in $(W',S')$ isomorphic to $(W,S)$. On the other hand, I am not aware of a more clever way to determine how $(W,S)$ embeds into $(W',S')$.