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Let $S$ be a Coxeter diagram, i.e. an unoriented graph, edges labelled by weights $3,4,5,...$ . This includes (affine) Dynkin diagrams,

Then

Theorem 3.1.3 of Brouwer and Haemers' Spectral Graph Theory says that the only graphs $\Gamma$ whose adjacency matrix $A_\Gamma$ has largest eigenvalue (less than) $2$ are the affine (resp. ordinary) ADE diagrams.

Now, the Coxeter group $W_S$ of the diagram always acts on the vector space $\text{Fun}(S)\simeq\mathbf{C}^{|S|}$ of functions on the graph. Does this action commute with taking $A_S$, and if so do we know anything about how the $A_S$-eigenspaces decompose as representations of $W_S$?

Or more generally - is there any other interesting connection between Coxeter groups/root systems/etc. and eigenvalues of the Laplacian? The purpose of this question is to understand if the above Theorem is a cute coincidence or whether there is more structure here to understand.

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    $\begingroup$ If you Google the keywords “Vinberg subadditive Dynkin diagrams” you should get info about what you want. Basically the affine ADE Dynkin diagrams are the unique graphs admitting labeling of their vertices by positive integers so that each vertex gets half the sum of its neighbors, and these functions (normalized to have smallest value one) tell you how to write the longest root of the corresponding root system (when we delete the affine node) in the basis of fundamental weights. $\endgroup$
    – Sam Hopkins
    Commented Jul 20 at 18:49
  • $\begingroup$ Argh, let me get this right: those coefficients I mentioned are the coefficients of the longest root in the basis of the simple roots, not fundamental weights. (For the fundamental weight coefficients you look at which nodes are connected to the affine node, but anyways that’s a different question…) $\endgroup$
    – Sam Hopkins
    Commented Jul 20 at 19:24
  • $\begingroup$ Are you sure you mean “graph Laplacian” and not “adjacency matrix”? The eigenvalues of the Laplacian of e.g. the 3 cycle are 3,3,0, at least the way I understand what is meant by graph Laplacians. $\endgroup$
    – Sam Hopkins
    Commented Jul 20 at 20:04
  • $\begingroup$ See also this previous MO question, of which yours is essentially a duplicate: mathoverflow.net/questions/468/… $\endgroup$
    – Sam Hopkins
    Commented Jul 20 at 20:15
  • $\begingroup$ See also these notes from a class taught by Alex Postnikov a while ago: math.mit.edu/~apost/courses/18.204_2018/DynkinDiagrams.pdf $\endgroup$
    – Sam Hopkins
    Commented Jul 20 at 20:16

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