Timeline for Spectra of Coxeter diagrams and representations of Coxeter groups
Current License: CC BY-SA 4.0
21 events
| when toggle format | what | by | license | comment | |
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| Jul 22 at 21:18 | comment | added | Pulcinella | @SamHopkins Thanks for this - I might comment again/edit my question after I do that. | |
| Jul 22 at 13:32 | comment | added | Sam Hopkins | I think you should read the notes of the Alex Postnikov class I linked to above. It explains how the simple reflections, in the usual Tits representation of a simply laced Coxeter group, correspond to the rows of the matrix $2-A_S$, which is why the largest eigenvalue being less than 2 is important for the finiteness of the group we get from these reflections. | |
| Jul 22 at 13:05 | comment | added | Pulcinella | Whether that operator is $W_S$-linear | |
| Jul 22 at 12:53 | comment | added | Sam Hopkins | What do you mean by “commutes with taking $A_S$”? | |
| Jul 22 at 12:41 | comment | added | Pulcinella | @SamHopkins Does $W_S$ act on $\text{Fun}(S)$, does it commute with $A_S$, and if so is there anything we can say about what $A_S$-eigenvalues the irreducible pieces of the representations has? Secondly, I am asking if there are any similar relations between Coxeter groups and spectral graph theory? | |
| Jul 22 at 12:38 | history | edited | Pulcinella | CC BY-SA 4.0 |
added 2 characters in body
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| Jul 22 at 9:27 | comment | added | Sam Hopkins | Okay but for arbitrary Coxeter diagrams, what is it that you are asking? As I mentioned I think I explained the meaning of the theorem, which I thought was the thrust of your question. Incidentally, you should probably correct the statement of the theorem to be about adjacency matrix eigenvalues, not Laplacian eigenvalues. | |
| Jul 22 at 9:20 | comment | added | Pulcinella | @SamHopkins Maybe I should have emphasised - I'm asking about arbitrary Coxeter diagrams (i.e. literally any finite undirected graph with edge numberings $3,4,...$, not just extended Dynkin) as opposed to arbitrary Dynkin diagrams. I'm happy to consider cases where $W_S$ is neither finite nor affine ADE. | |
| Jul 22 at 9:07 | comment | added | Sam Hopkins | In particular regarding “The purpose of this question is to understand if the above Theorem is a cute coincidence” - I would say what I have explained shows it is not a coincidence. | |
| Jul 22 at 9:06 | comment | added | Sam Hopkins | @Pulcinella the way I see it, what the stuff I talked about explains is that this fact that the affine Dublin diagrams have eigenvalues less than or equal to two exactly corresponds to the fact that the finite Coxeter groups are classified by the finite Dynknin diagrams. In other words, if your diagram has an eigenvalue bigger than two, you will have infinitely many elements in your Coxeter group (or infinitely many roots, if you like). In that sense I think it does say something about “arbitrary Dynkin diagrams.” But maybe I misunderstood your question. | |
| Jul 22 at 8:37 | comment | added | Pulcinella | @SamHopkins Thanks for all your comments so far. I've looked at your links, and I'm not able to find anywhere that deals with arbitrary Coxeter diagrams, only with (affine) Dynkin ADE ones. Might you know where I could find this? I feel like to understand the Laplacian/root system relation I should understand this less nice/more general case. | |
| Jul 21 at 19:33 | comment | added | LSpice | @SamHopkins, re, I just meant it as a very simple-minded question: you suggested to Google "Vinberg subadditive Dynkin diagrams" for info, and I wasn't sure which of the hits might be providing the desired info. (I don't know this result of Vinberg's well enough to sort it out myself.) But I think that your further comment now serves as a specific reference. Thanks! | |
| Jul 20 at 21:34 | comment | added | Sam Hopkins | @LSpice: I’m sorry, I don’t understand your comment. You’re looking for somewhere what Vinberg showed is explained? See, e.g., section 2 of arxiv.org/abs/1704.05024 | |
| Jul 20 at 21:22 | comment | added | LSpice | @SamHopkins, re, it wasn't completely obvious to me which of the results that came up might be relevant. The first result, indirectly duplicated by many of the others, is Happel, Preiser, and Ringel - Vinberg’s characterization of Dynkin diagrams using subadditive functions with application to DTr-periodic modules. Is that the paper you had in mind? | |
| Jul 20 at 20:16 | comment | added | Sam Hopkins | See also these notes from a class taught by Alex Postnikov a while ago: math.mit.edu/~apost/courses/18.204_2018/DynkinDiagrams.pdf | |
| Jul 20 at 20:15 | comment | added | Sam Hopkins | See also this previous MO question, of which yours is essentially a duplicate: mathoverflow.net/questions/468/… | |
| Jul 20 at 20:04 | comment | added | Sam Hopkins | 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. | |
| Jul 20 at 19:24 | comment | added | Sam Hopkins | 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…) | |
| Jul 20 at 18:49 | comment | added | Sam Hopkins | 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. | |
| Jul 20 at 18:03 | history | edited | Pulcinella | CC BY-SA 4.0 |
edited title
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| Jul 20 at 17:56 | history | asked | Pulcinella | CC BY-SA 4.0 |