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Note #1: title of this post was modified as per a great suggestion from Todd Trimble

Note #2 ( to site moderators): I truly consider this question to be a legit "meta" question about the MathOverflow community - in particular, the general attitude of the MathOverflow community toward a particular tool which is available to them courtesy of HSM Coxeter. But Todd Trimble thinks not, so in deference to the opinion of an experienced MO member, I've reworded the title of the question as he suggested - see discussion below in comments.)

Here is the question I'd like to post to MO, if appropriate.


Background:

Please see my recent question at MathOverflow:

$E_6$, $E_8$, and Coxeter's (anti-)prismatic projections of the n-dimensional cross-polytopes

Question:

Assume for the sake of discussion that in the examples in the question above:

(a) projection of the vertices of the 8-space cross-polytope onto the vertices of the 3-space octagonal prism DOES preserve all important properties of the roots of $E_8$ (as expressed in their usual {128,112} coodinatization in 8-space)

(b) projection of the vertices of the 9-space cross-polytope onto the vertices of the 3-space nonagonal antiprism DOES preserve all important properties of the roots of $E_6$ (as expressed in 9-space in their usual {18,27,27} coordinatization)

Then under these assumptions, is there still some reason why you personally would NOT think it worthwhile to investigate the roots of $E_6$ and $E_8$ and their relations to one another using the visualizations of these roots in 3-space that can be obtained via Coxeter's projections?

If so, what would your reason be?

For example, would your reason simply arise from your own "personal mathematical aesthetic", e.g a distaste for working from "diagrams" rather than linear symbolic expressions? (I mention this possibility because Dr. (Sam) Eilenberg once chewed me out for doing a 140-page proof about certain properties of the "derivation trees" (ordered finite rooted directed trees with labelled nodes) of derivations in context-free grammars - he told me that in his opinion as an automata-theorist, nothing worthwhile could ever be learned from the properties of "diagrams".)

Or is there some other reason you personally wouldn't use such projections?

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    $\begingroup$ I see that you've been registered as a user for only a couple of days, so you may not be fully aware how things work around here. MO meta is to be used for questions regarding actions and policies of MathOverflow as a StackExchange site. It is for example legitimate to inquire why a question was closed, i.e., what general MO guidelines led to a decision to close. (We actually have a dedicated thread for reopening inquiries.) But taking an opinion poll on whether a certain form of research investigation would be mathematically worthwhile seems to me to be off-topic for this site. $\endgroup$ – Todd Trimble Dec 10 '17 at 14:11
  • $\begingroup$ @ToddTrimble - thanks for taking the time to give me some guidance on this, Todd - really appreciate it. I will leave it to the discretion of the moderators as to whether this is or isn't a legit queston "about the MathOverflow community". If not, perhaps the wording of the baseline instruction description should be changed ???? $\endgroup$ – David Halitsky Dec 10 '17 at 14:21
  • $\begingroup$ Some form of the question that you've asked here might work at MathOverflow; as I see it, it's more of a mathematical question than a site policy question. But I anticipate many would see it as borderline appropriate there, so it would probably have to be done carefully. I think if you tweaked the question above to: "would the following inquiry be appropriate for MO?", then it would be fully appropriate for MO meta, and you might get some good ideas on how to turn your poll into a good MO question. $\endgroup$ – Todd Trimble Dec 10 '17 at 14:33
  • $\begingroup$ @ToddTrimble - thanks for the suggestion ! Will do !!! $\endgroup$ – David Halitsky Dec 10 '17 at 14:38
  • $\begingroup$ "I will leave it to the discretion of the moderators as to whether this is or isn't a legit queston...." It's not up to the discretion of the moderators. Any user who has enough points, moderator or not, can choose to vote to close this question. Further, if there are enough votes to close, then users can vote to delete. $\endgroup$ – Gerry Myerson Dec 10 '17 at 22:30
  • $\begingroup$ @GerryMyerson - thanks as always for the clarification, Gerry. But please note that I did take Todd's advice and "tweak" the title of the question . . . so perhaps the issue won't even arise . . . $\endgroup$ – David Halitsky Dec 10 '17 at 22:37
  • $\begingroup$ @GerhardPaseman - I would like to ask if you consider this question of "general interest": math.stackexchange.com/questions/2568623/…. I ask this question because if you agree that this question is of general interest, it is difficult for me to see how you can at the same time claim that the question here is NOT of general interest. If you don't have time to respond, please don't feel obliged. Just wanted to bring this to your attention. $\endgroup$ – David Halitsky Dec 16 '17 at 0:50
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It is possible that this question is useful to you. (I could put up an argument that it is not, but you might counter with additional detail that would render my argument invalid.). However, I see the question (as I understand it) to be too narrow as to be useful to anyone else.

When simplified and abstracted, my take is that you are asking "Given some not easily testable assumptions (so that obvious reasons for not studying do not apply), are there any reasons you can think of for not studying this particular research problem?". If there were many researchers on this planet who were interested in your topic at hand, this question and its answers might serve them, especially given unmentioned specifics like time, place, and politics which might hold for them.

Now for my assumption. I assume the question and its answers will apply only to you, and not to future readers of the forum, because it is too specific and it asks for opinion. I also assume that (unmentioned) conditions that prompt the question are somewhat temporary and unique to your situation, and that these conditions might render the answers inapplicable to later readers who might find themselves in an even remotely similar position. I am not seeing the question above as a good fit for MathOverflow because the forum is more for matters that are verifiable and not subject as much to discussion, advice, or opinion (even though there are some advice questions on the forum).

If you really need advice, make it a GOOD advice question so that others might make use of the answers. For example, a reasonable question might be "How can I cope with the situation where an advisor tells me to abandon my current research topic and switch to one that interests me less? By the way, my specific situation is, I am studying X, advisor is telling me to switch because of reason R, and I think that S means there is something worth spending time with.". This is an example where answers of both kinds can help. The general answers can talk about why (or why not) you should listen to the advisor and not give pushback, or how to work on two problems at the same time, or the risks you take in following your own (possibly unproductive) track. The subject matter experts can give specific answers about the validity of your assumptions and whether your approach has been tried already.

Some of the above is more appropriate for academia.stackexchange, and some is stuff you should ask a mentor or someone else who knows you and the situation better; we can give war stories and suggestions which may be bad for you because we do not have (and can never get) the whole picture.

I recommend talking to people face to face about this. If you have to ask on this forum, ask in a general enough way so that more people can get use from the question and possible answers.

Gerhard "How To Fix Published Errors" Paseman, 2017.12.10.

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  • $\begingroup$ Although I make an implicit assumption that the situation prompting this is a graduate student versus advisor situation, the reasons apply to other dynamics as well even if the specifics don't . So imagine also versions of my response tailored to research team, or lab director versus funding organization. Gerhard "Let's Make It Widely Applicable" Paseman, 2017.12.10. $\endgroup$ – Gerhard Paseman Dec 10 '17 at 16:50
  • $\begingroup$ thanks so much for that thoughtful response. I think you may have a valid point that the question would be too narrow and therefore not interesting to many. But I do disagree about your characterization of the question - I am not at all asking whether it's worth studying a particular research problem. Rather, I am asking why mathematicians might not consider it uself to use a particular tool that might be applicable in many different research problems, not just mine. I any event, thanks so much again for your response. $\endgroup$ – David Halitsky Dec 10 '17 at 17:03

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