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Given a Coxeter system $(W,S)$ with reflections $T$, and any subset $A \subseteq T$, it is known that the reflection subgroup $W_A$ generated by $A$ has a canonical choice $S_A$ of generators so that $(W_A,S_A)$ is a Coxeter system (a result of Deodhar and of Dyer).

Question. Given two Coxeter graphs $G, G'$, is there some way to tell whether the Coxeter group $W$ associated to $G$ has a reflection subgroup $W'$ with Coxeter graph $G'$? I suspect this is hard in general, but are there nontrivial necessary or sufficient conditions known? In particular, is there some general principle by which I could recognize that the Coxeter group of type $B_n$ has a reflection subgroup of type $D_n$ just from the graphs?

Clearly $G'$ being an induced subgraph of $G$ is sufficient, but not necessary. I'm hoping there are more interesting things that can be said in general.

Another question asks for a stronger thing (an algorithm finding all "Coxeter subgroups"; as far as I can tell, that asker does not even require them to be reflection subgroups).

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    $\begingroup$ The B/D example is explained by Borel-de Siebenthal theory, I believe. $\endgroup$
    – Sam Hopkins
    Commented Jan 17, 2020 at 21:37
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    $\begingroup$ (Borel-de Siebenthal theory says that sub-root systems of maximal rank of a given root system correspond to Dynkin diagrams obtained from the affine Dynkin diagram corresponding to the original root system by deleting a single node.) $\endgroup$
    – Sam Hopkins
    Commented Jan 17, 2020 at 21:39
  • $\begingroup$ As an extension of the process described in Sam's comment, Dyer gives a description of the reflection subgroups in affine type. The Coxeter graphs of reflection subgroups are those which can be constructed from the original affine Coxeter diagram by successively removing vertices and converting finite type diagrams to the corresponding affine diagrams as desired. See the end of this article. $\endgroup$
    – Grant B.
    Commented Jan 18, 2020 at 7:04
  • $\begingroup$ Though in general (i.e., outside affine type) we will have infinite rank reflection subgroups (see, for instance, David Speyer's answer here). So a general algorithm of the form "convert one type of diagram to another and delete points" would need to address this. $\endgroup$
    – Grant B.
    Commented Jan 18, 2020 at 7:19
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    $\begingroup$ @ChristianGaetz: For right-angled Coxeter groups, the article arxiv:1910.04230 might interest you. It can be deduced from it an algorithm which determines whether or not a right-angled Coxeter group $C(\Gamma_1)$ embeds as a reflection subgroup into another right-angled Coxeter group $C(\Gamma_2)$. $\endgroup$
    – AGenevois
    Commented Jan 18, 2020 at 10:00

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