Talk:Semisimple representation

The following is an archived discussion of the DYK nomination of the article below. Please do not modify this page. Subsequent comments should be made on the appropriate discussion page (such as this nomination's talk page, the article's talk page or Wikipedia talk:Did you know), unless there is consensus to re-open the discussion at this page. No further edits should be made to this page.

The result was: promoted by Montanabw^(talk) 17:11, 3 January 2020 (UTC)

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• ... that how a spinning object influences the rotation of another spinning object in quantum mechanics is described by the structure of certain semisimple representations? Source: Clebsch–Gordan coefficients come from angular momentum coupling as evident from the WP article and its sources; the relation to semisimple representations comes from: "The representations of some semisimple and reductive Lie groups have become of increased importance in physics... to study CGCs of semisimple Lie groups." from Klimyk, A. U.; Gavrilik, A. M. (1979). "Representation matrix elements and Clebsch–Gordan coefficients of the semisimple Lie groups". Journal of Mathematical Physics. 20 (1624). doi:10.1063/1.524268.
  • ALT1:... that Clebsch–Gordan coefficients, which describe how spinning objects influence each other in quantum mechanics, come from the structure of semisimple representations? Source: same as above
  • ALT2:... that adding together two simple things might only make a semisimple one? Humor hook; Source: definition, e.g. Ch. 6, § 1.1, Definition 1 (ii) of Claudio Procesi (2007) Lie Groups: an approach through invariants and representation, Springer, ISBN 9780387260402.
  • ALT3:... that simple ⊕ simple = semisimple? Humor hook; Source: same as above; we could also use "+" instead of "⊕"

• Reviewed: Exempt - 4 DYKs

Created by TakuyaMurata (talk) and MarkH21 (talk). Nominated by MarkH21 (talk) at 12:46, 15 November 2019 (UTC).

• Comment (not a review): Don't the hidden-by-default proofs violate MOS:DONTHIDE "Collapsible templates should not conceal article content by default upon page loading"? —David Eppstein (talk) 02:03, 17 November 2019 (UTC)

• Ah, you’re right. Fixed and un-autocollapsed. — MarkH21 (talk) 02:11, 17 November 2019 (UTC)

• More not-really-reviewing: the entire "Examples and non-examples" section is unsourced. According to the DYK rules, every paragraph that is not merely a summary of later material (or plot summary of fiction) needs a source. —David Eppstein (talk) 01:41, 18 November 2019 (UTC)

• Pinging TakuyaMurata, did you have a source that we can cite for the non-example that you added? — MarkH21 (talk) 03:04, 18 November 2019 (UTC)

• I have added a few (not saying "enough"). -- Taku (talk) 23:40, 18 November 2019 (UTC)

• Full review needed. BlueMoonset (talk) 06:59, 24 November 2019‎ (UTC)

• On the day it was nominated, it was new enough and long enough. According to QPQ check, @TakuyaMurata: does not need a QPQ. Earwig is okay with it. Alt0 is the best hook IMHO, but it could be tightened up. I don't understand the math, so having a mathmetician look at it would be good. --evrik ^(talk) 21:50, 18 December 2019 (UTC)

• Requesting a second opinion on the math in the article. BlueMoonset (talk) 19:27, 20 December 2019 (UTC)

• I'll post a request for review in WikiProject Math because... there's a lot. Kingsif (talk) 17:16, 21 December 2019 (UTC)

• The math in the article looks correct. XOR'easter (talk) 21:37, 26 December 2019 (UTC)

• Thanks XOReaster - from previous reviews and this affirmation, I've looked at the article and all the current hooks look fine. I think Alt1 is easier to understand than alt0, but they're all fine. Kingsif (talk) 23:07, 26 December 2019 (UTC)

• I like alt1 a little more than alt0. XOR'easter (talk) 00:10, 27 December 2019 (UTC)

• @Kingsif: where does the article mention Clebsch–Gordan coefficients? Yoninah (talk) 23:07, 28 December 2019 (UTC)

• @Yoninah: From my understanding of the sources used for the first hooks, it's the sentence By Weyl's theorem on complete reducibility, every finite-dimensional representation of a semisimple Lie algebra over a field of characteristic zero is semisimple. Of course, neither Clebsch–Gordan coefficients nor angular momentum coupling are mentioned in the article, and I now see that the Klimyk/Gavrilik source isn't used in it to even support some complex mathematics that could be giving the same meaning. @MarkH21 and XOR'easter: to ask if the hook can be explicitly added to the article or something? Kingsif (talk) 23:31, 28 December 2019 (UTC)

• @Yoninah and Kingsif: That’s not a requirement for DYK to my understanding right? We certainly could add the sourced statement from the hook to the article, but then we would have to add a lot of other applications to the article though, since this would be undue prominence for Clebsch–Gordan coefficients. Semisimple representations are used all across physics! This is just one interesting application. — MarkH₂₁^talk 23:35, 28 December 2019 (UTC)

• @MarkH21: please read WP:DYK#Eligibility criteria 3. Cited hook. Yoninah (talk) 23:39, 28 December 2019 (UTC)

• @Yoninah: Ah my bad, I don't know why I assumed that it wasn't an explicit rule. Will add to the article with an "Applications" section soon. — MarkH₂₁^talk 23:42, 28 December 2019 (UTC)

• @Yoninah: Wasn't Alt0 cited? --evrik ^(talk) 20:05, 2 January 2020 (UTC)

• @Evrik: I didn't check. The reviewer preferred ALT1 for simplicity, and I agree, so that's the one I checked. Yoninah (talk) 21:24, 2 January 2020 (UTC)

• I would have passed this on December 18 if I had better understood the math. I am approving Alt0, as I believe it is sourced and ready to go. --evrik ^(talk) 21:35, 2 January 2020 (UTC)

• @Evrik and Yoninah: The source for ALT0 and ALT1 are the same, they're just minor variations of the same hook. I've added a basic "Applications" section to the article mentioning the fact in the hooks, this will be expanded in the future but should be sufficient for the DYK. — MarkH₂₁^talk 21:40, 2 January 2020 (UTC)

• @Evrik: thank you for your interest, but I was in the middle of a discussion with the nominator and I am not ready to sign off until I review his new edits. The ALT0 hook is too long and multisyllabic for comfortable reading and there's no reason to rush this off to the main page. I'm looking at MarkH21's work now. Yoninah (talk) 22:20, 2 January 2020 (UTC)

• Yoninah, FWIW I did the first review on this and I think I used this for QPQ credit. Just saying ... in any case I glad to see this move forward. --evrik ^(talk) 16:09, 3 January 2020 (UTC)

• Thank you, MarkH21. ALT1 is now sourced and the offline ref is AGF and cited inline. Rest of review per evrik and Kingsif. ALT1 good to go. Yoninah (talk) 22:27, 2 January 2020 (UTC)

Wikipedia:Miscellany for deletion/Draft:Semisimple representation

The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

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Any editor nominating this for Deletion, I attempted to have this page deleted under MFD and the author of the page moved it to mainspace during the MFD. Per accepted convention I withdrew the MFD, however I would like to firmly register that I oppose any return to Draftspace without the stick of CSD:G13 being in place including a no REFUND allowed rule. Either the page survives in mainspace, it is returned to Draft namespace and it stays of the G13 rail, or it gets deleted without any allowance for WP:REFUND. Hasteur (talk) 01:09, 9 November 2019 (UTC)

No other editor suggested such a restriction on this page, and it cannot therefore be considered to have been imposed by consensus. I do not see any reason why this should be returned to draft status, but if it should be, it will be in precisely the same position as any other draft. The availability of refund is a general policy and cannot be foreclosed in advance by a singel editor, although an admin asked for a future REFUND on this page might consider this history. DES ^(talk)_{DESiegel Contribs} 03:39, 9 November 2019 (UTC)

The discussion above is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

General definition of a semisimple rep

To respond to the edit summary “the more general semisimple objects don't really belong in the first sentence, ” by @MarkH21:, first I don’t disagree. But it’s still important, I think, to mention semisimple module as well as semisimple operator are examples of semisimple representations in the lead; perhaps in the second paragraph. (Incidentally, the opposite is also possible in the sense: given a set of operators, one can consider the algebra generated by it and then a semisimple representation is also a semisimple module over that algebra.) I wonder what the best way is. —— Taku (talk) 21:13, 15 November 2019 (UTC)

@TakuyaMurata: Sure, at the very least we can create a section in the article body and then summarize that later in the lead. — MarkH21 (talk) 10:58, 16 November 2019 (UTC)

The thing is this article is about a specific type of a representation (as opposed to a representation in general) and so it seems strange to have a section that discusses the definition of a representation in a somehow generalized sense. Anyway, for now, I will add a few sentences to the "equivalent characterization" section; note the proof is abstract in that it doesn't care a representation is of a group, an algebra or a single operator. -- Taku (talk) 23:53, 17 November 2019 (UTC)

I think that having different sections for different kinds of representations is fine. For instance, adding to the existing sections on groups and Lie algebras. — MarkH21 (talk) 23:56, 17 November 2019 (UTC)

No, I am *against* such a structure. Because this article is about a rep not an algebraic object, the emphasis should be on the structure of a rep. For example, the article needs (and I am planning to add later) a discussion on multiplicity of simple reps; that's independent of whether one is considering a rep of a group or an algebra. (In fact, with due care, one can more ore less identify a rep of a Lie group and a rep of a Lie algebra.) -- Taku (talk) 01:02, 18 November 2019 (UTC)

Ah sorry, I misunderstood what you meant. But representations of algebras and operators are representations; should this just be about group representations? It wouldn’t be terrible to include a generalization section even if it was focused on group representations, particularly since “semisimple representation” may refer to them in literature (I also don’t mean to include semisimple objects in general). — MarkH21 (talk) 01:23, 18 November 2019 (UTC)

I am not sure if I follow. I am against structuring articles according to types of algebraic objects. In physics, *as far as I understand*, it's common to treat the same rep as a that of a Lie group and that of a Lie algebra. In other words, the focus is on a rep and you stay flexible about whether the action is through an algebra or a group. This is why it's a bad idea to have an article structure based on algebraic types (am I making sense?) -- Taku (talk) 22:56, 18 November 2019 (UTC)

I am for having a separate section on the infinite-dimensional rep case, since in that case, one usually has the *completion* of a semisimple representation, as opposed to a semisimple representation, as you have pointed out, (and so one needs an extra care). -- Taku (talk) 23:27, 18 November 2019 (UTC)

On the last point: another option is to have a section of a regular representation (which is semisimple rep or the completion of a semisimple rep in the interesting case). Maybe we don’t need to give a detailed discussion of an infinite-dimensional semisimple rep. —- Taku (talk) 03:25, 19 November 2019 (UTC)

Merge proposal

@TakuyaMurata: What is your justification for merging Isotypic component to this article? –LaundryPizza03 (dc̄) 01:35, 11 July 2024 (UTC)

Thank you for asking. The rationale for the merger proposal is that the article isotypic component concerns about a particular instance of an isotypic component. This article also discusses an isotypic component but in a more general context and merging the former into this one should be helpful to the readers. —- Taku (talk) 05:09, 11 July 2024 (UTC)

I haven't taken my own look, but I guess that is okay. –LaundryPizza03 (dc̄) 06:12, 11 July 2024 (UTC)