Talk:Double group/Archive 1

Archive 1

Mathematical translation

From the appendix in Albert Cotton's "Chemical applications of group theory," my impression is that the "double groups" concern the character theory of the double covers of the finite subgroups of the SO(3), so finite subgroups of SU(2). These are the binary subgroups, such as the binary octahedral group S₄*, binary icosahedral group A₅* etc, and are often denoted by an asterisk. These are special cases of character and spin character tables determined by Frobenius and Schur for symmetric and alternating groups, with well-known general combinatorial rules (hook length formulas, staircase rules, etc, summarised in the text books of G. de B. Robinson, Gordon James, Fulton & Harris, etc). The labelling by Coxeter-Dynkin diagrams is known from the McKay correspondence, with the diagrams showing the rule for tensoring with the 2-dimensional representation, see [1]. Mathsci (talk) 14:41, 17 March 2022 (UTC)

Kudos

You've all done a great job of turning this page into a nicely informative page. Kudos to all! KeeYou Flib (talk) 19:28, 25 March 2022 (UTC)

Three deletions

1. Reference to the article by Bethe. I am unable to verify whether the cited article contains material relevant to double groups or not.

2. Section "Character tables of crystallographic point groups" deleted. This is completely irrelevant to the topic of the article.

3. Section "McKay correspondence" Also deleted as irrelevant.

There is nothing wrong with the deleted sections 2 and 3, but they are misplaced. They make no mention of the double groups.

As far as I know the only application is in relation the theory of magnetism in molecules, specifically octahedral and square-planar complexes of copper(II).

Rather than deleting whole sections, let's talk. I agree that these sections are extremely unclear as to what they have to do with double groups. In order for them to stay, they need improvement. KeeYou Flib (talk) 16:11, 28 March 2022 (UTC)

I have added tags to these sections asking for clarification. Hopefully someone will respond? KeeYou Flib (talk) 20:45, 28 March 2022 (UTC)

Could Petergans please sign his posts? An AfD was started for "Double group" by D.Lazard. Various users contributed to that discussion, but Petergans seems not to have acknowledged that these terms are well understood in physics and mathematics. That was explained carefully in the AfD discussion, where, apart from Petergans, there was WP:consensus.

The conclusion of the AfD was not that Petergans could create arbitrary content on Double group. Four mainstream text books in physics and one in mathematics appear in the reference section and describe in great detail double groups (character tables of crystallographic point groups and the double groups). SO(3) and its double cover SU(2) are discussed in undergraduate courses (some using unit quaternions). Their representation theory is part of second year quantum mechanics and angular momentum, including spin representations, character formulas and Clebsch-Gordon rules; similarly the character theory of finite groups is taught in undergraduate mathematics courses. Cornwell's "Group theory in physics. Vol. 1", and the four other books, covers both the theory and its applications, sometimes in a pedestrian way. The finite groups studied in physics, material sciences, chemistry, etc are subject to crystallographic restriction rules (possible orders of group elements are 1, 2, 3, 4 and 6).

Petergans' edits are disruptive because he has attempted to delete content and references to text books on physics and mathematics that unambiguously discuss double groups; and because so far his edits show no coherent sentence about "double groups" in the lead. Declaring that these references and subject matter are irrelevant is hard to understand in scientific terms. Until fairly recently, there was mathematical expertise on this topic at the University of Leeds (quivers and the MacKay correspondence, see here); given that all those text books are readily available in several of the university's libraries and online on bookos, why have all these text books and their content been deemed to be irrelevant? On wikipedia, the standard process, when editors question the relevance of references, is to list those references on WP:RSN—in this case the five text books and Bethe's 1929 article in German and English. Given the AfD and Petergans' deletions, there is a possibility that the article will be either speedily deleted or subjected to a further AfD. User:Qflib did not participate in that AfD discussion. Mathsci (talk) 14:23, 29 March 2022 (UTC)

My own criticism - and I think it is valid - is that the connections are not self-evident to a general reader. In other words, what you've written is not sufficiently accessible. If you would simply provide a sentence or two in each section making it clear to a general reader that these two sections are directly connected to the topic at hand, it would solve the problem completely in my opinion. KeeYou Flib (talk) 17:26, 29 March 2022 (UTC)

This discussion about Richard Feynman comes to mind whenever it's my task to explain a complex topic. https://en.wikiquote.org/wiki/Talk:Richard_Feynman#Teaching_quote KeeYou Flib (talk) 17:46, 29 March 2022 (UTC)

That is just false and the problem is with the lead, in its current version. As explained in the AfD (please read re-read it again), Petergans' sub-stub was incomprehensible and it still is. He hasn't made any effort to present as they appear in those physics text books. So it's useless to physicists or mathematicians who find out what "double group" mean, and why the word "double" appears. Cotton's book just refers to the accounts of Koster and Bethe, so does not even briefly summarise that theory. The phrase "double groups are used in an extension of group theory" is not explained at all. On the other hand, the five references give completely accessible explanations of double groups and why the word "double" appears. Indeed it concerns finite subgroups Γ of SO(3) which naturally lead to subgroups Γ* of SU(2), with double the order: thus a central extension of Γ by {±I}) and the double cover SU(2) → SO(3) induces a double cover Γ* → Γ. This is standard language in mathematical physics (cf the classic books of Wigner and Weyl). I've given that basic definition: the WP:ONUS now is on you to navigate your way through that straightforward material.

After all, wikipedia is about finding good sources and then summarising them so that others can verify the content. That applies to any article (e.g. my recent edits to Henry VII of England and material on John Cabot). Some experience and knowledge is useful. In my case, in the AfD I referred to an Inventiones Mathematicae article; in that article the character tables of the double groups of S₄ and A₅ are written down.

Perhaps it's easiest if I ask four questions. (a) Do you know what S₄ might be? (b) do you know what A₅ might be? (c) Do you know how they appear as symmetries of polyhedrons? (d) Do you know about the double cover SU(2) → SO(3) and how it fits in with quaternions? That is the first step to understanding the material. From the text books, double groups are used in various disciplines such as mineralogy, crystals, material sciences, Brillouin zones, etc. Mathsci (talk) 20:47, 29 March 2022 (UTC)

Of course you’re welcome to your opinion that the article is crystal clear as is. I suggested that it’s not, and you reject that. We seem to be at an impasse. Someone else will have to resolve matters, I suppose. KeeYou Flib (talk) 02:14, 30 March 2022 (UTC)

I decided to boldly edit the page, while leaving "your sections" strictly alone. Hopefully things are somewhat improved. KeeYou Flib (talk) 14:13, 30 March 2022 (UTC)

I do have to say that for the record, I did in fact participate in the AfD discussion. Look for "KeeYou Flib", not "Qflib". KeeYou Flib (talk) 21:03, 3 April 2022 (UTC)

New Introduction

I have the intention of developing a new introduction, so the new layout is a move in the right direction. However, it is much too technical as it is, to the point of being incomprehensible to non-mathematicians. My idea was to give a simple introduction to the mathematical development of point group theory and character tables: point group symmetry operations are, in this context, to be viewed as a sub-set those of corresponding double groups. The current text will be used as a basis for development. Petergans (talk) 19:31, 30 March 2022 (UTC)

Sounds good. Let's all three of us work to achieve Wikipedia:Consensus. KeeYou Flib (talk) 13:38, 1 April 2022 (UTC)

I have now completed the development. In the new version everything that is not concerned specifically with double groups has been excluded. To compensate for this all relevant concepts are now linked to appropriate WP articles. Petergans (talk) 10:49, 2 April 2022 (UTC)

Direct product

Can one say that a double group is the direct product of a molecular symmetry group with a group of order 2 consisting of rotation by 2π and the identity operator? Dirac66 (talk) 15:12, 1 April 2022 (UTC)

I've not seen this stated explicitly anywhere. I guess that the term "double group" originates from the fact that these groups have two distinct identity operations. The order of the double group is only one more than the order of the point group. Nevertheless, I agree with your conjecture. Are you able to do check whether double groups were included in Bethe's original paper? If not, where did the concept originate? Petergans (talk) 10:37, 2 April 2022 (UTC)

This reference may be helpful, in particular chapter 14. https://wiki.physics.udel.edu/wiki_qttg/images/b/b5/DRESSELHAUS%3Dgroup_theory.pdf KeeYou Flib (talk) 12:18, 3 April 2022 (UTC)

Thank you for this pdf. It provides a clearer presentation of the character table for the double group O, in which col. 4 is labelled 3C₄² + 3RC₄², etc.. The labelling of the columns is not explained in Cotton's book, from which I copied the table, so I will augment the labelling in the article. It will also make clear that the double group is indeed the direct product as suggested by user:Dirac66 Petergans (talk) 14:31, 3 April 2022 (UTC)

• I have reverted recent edits as they do not improve the article and are completely misleading; those edits misinterpret wholly the recent conclusions of the WP:AfD. The edits removed removed all text books "double groups": they were purely disruptive so have been reverted. Changes to articles are made by wp:consensus and incrementally. Using a sandbox to create a preferred version of an article is an example of wp:own, contrary to wikipedia policy. Given the repeated refusal to acknowledge the outcome of the AfD and the continued removal of sources, I am asking the administrator User:Johnuniq to help manage the continued disruption on this page. Mathsci (talk) 16:12, 3 April 2022 (UTC)

Let me be clear: The deleted sections relate to group theory in general and, as such, there was nothing wrong with them. But, in this context, they are irrelevant, saying nothing at all about the subject defined by the article title. I could not find anything about the deleted topics in any source dealing with double groups that I consulted. The content of the deleted sections is well covered by other articles on group theory.

To justify their inclusion of the deleted sections, some evidence must be presented that shows their direct relevance to the topic.

Please note that the wholesale reversion removed valid material that had been added, such as the example group table. Petergans (talk) 19:52, 3 April 2022 (UTC)

Please don't refactor things in this way, Petergans. Your edits have been disruptive today so have been restored to a more stable version. I hope that User:Johnuniq, an administrator, will intervene. The references are the following:

1. Bethe, Hans (1929). "Termaufspaltung in Kristallen" [Splitting of Terms in Crystals]. Ann. Physik (in German). 395 (3): 133–206.
2. Bethe, Hans (1996). Selected Works of Hans A. Bethe with commentary. Translated by Hans Bethe. World Scientific. pp. 1–72. ISBN 9789810228767.
3. Murnaghan, Francis D. (1938). The theory of group representations. Baltimore: Johns Hopkins Press.
4. Koster, George F.; Dimmock, John O.; Wheeler, Robert G.; Statz, Hermann (1963). Properties of the thirty-two point groups. Cambridge, Mass.: The M.I.T. Press. MR 0159664.
5. Miller, Willard, Jr. (1972). Symmetry groups and their applications. Pure and Applied Mathematics. Vol. 50. New York-London: Academic Press. MR 0338286.{{cite book}}: CS1 maint: multiple names: authors list (link)
6. Burns, Gerald (1977). Introduction to group theory with applications. Materials Science and Technology. New York-London: Academic Press. ISBN 0-12-145750-8. MR 0482722.
7. Cornwell, J. F. (1984). Group theory in physics. Vol. I. Techniques of Physics. Vol. 7. Academic Press. ISBN 0-12-189801-6. MR 0751778.

These are Bethe's original paper (and translation) as well as the standard text books that treat "double group" in a self-contained, accessible and encyclopedic way. Most are written from a physics perspective. Petergans has written several times that these sources are irrelevant. On the other hand, initially D.Lazard nominated Wikipedia:Articles for deletion/Double group because as initially presented as a sub-stub, it was not comprehensible. Later the discussion at the AfD gave evidence that there were notable references treating the subject and that it was actually known theory amongst physicists (and mathematicians). On the other hand there was no wp:consensus for Petergans' edits attempting to restructure Group (mathematics).

Physicists' writings on "double groups" are easily accessible—Hans Bethe was a nuclear physicist and his original 1929 paper (now translated into English) adopts the same writing style as his contemporaries Wigner, Weyl, von Neumann, etc. During the AfD, I used mathscinet to check WP:RSs for material on "double group"; I know the material well from the book of Murnaghan (and elsewhere). Deleting the seven references above from the article cannot be justified in any way. For each of the seven cases, the topic of "double groups" occurs as a significant application of "group theory in physics". Mathsci (talk) 23:01, 3 April 2022 (UTC)

McKay correspondence

There is no evidence that this section has any relevance specific to the subject of double groups. I propose it be deleted unless some evidence is presented. Petergans (talk) 08:18, 4 April 2022 (UTC)

The binary subgroups Γ* ⊂ SU(2) are double groups that arise as finite subgroups Γ ⊂ SO(3). They are associated with regular polytopes (as explained in the book of H. S. M. Coxeter). The character tables appear in the McKay correspondence, but the ADE classification dates back much earlier (probably to Élie Cartan). For applications in physics or chemistry, there is a crystallographic restriction which dictates that the order of each group element is 1, 2, 3, 4 or 6.

Previously Petergans has written multiple times that the references to double groups appearing in Bethe's original article and subsequent text books are irrelevant: that appears to have been some kind f "culture gap" between physics and chemistry. The same "culture gap" seems to be in operation here, in this case between mathematics, prompted by aspects of string theory, and chemistry. If we return to the seminal 1929 article in Annalen der Physik (and its English translation), on page 16 Hans Bethe writes "We shall designate the group so obtalned, which has twice as many elements as the original group, as the crystal double group, and shall inquire about its irreducible representations. This accomplishes our purpose of representing every element of the simple group by two matrices (a two-valued matrix). This procedure corresponds somewhat to the construction of the Riemann surface for the study of multiple-valued functions." On wikipedia, there are articles binary octahedral group, binary tetrahedral group and binary icosahedral group (all double groups). The subgroups ${\mathfrak {A}}_{4}$, ${\mathfrak {S}}_{4}$, ${\mathfrak {A}}_{5}$ ⊂ SO(3) have their corresponding binary subgroups ${\mathfrak {A}}_{4}^{*}$, ${\mathfrak {S}}_{4}^{*}$, ${\mathfrak {A}}_{5}^{*}$. Bethe discusses the quantum mechanical wave functions and tensor products of irreducible representations, citing Wigner & von Neumann (1928). F. Albert Cotton reproduces the character table of the icosahedral group, even though it is not a crystallographic point group (having 5-cycles (1,2,3,4,5) so elements of order 5 in ${\mathfrak {A}}_{5}$).

The representation theory and character tables of double/binary subgroups were explained in full detail by Francis D. Murnaghan in 1938, citing Bethe's 1929 paper. There is also a chapter on "crystallographic groups" explaining the crystallographic restriction (page 337). Tensor product rules are also described. Petergans has repeatedly claimed that this source is irrelevant.

Why there is no mention of physics and quantum mechanics is a mystery. The McKay correspondence (with references) is described in the article of Bertram Kostant in the volume "The mathematical heritage of Elie Cartan". Mathsci (talk) 15:18, 4 April 2022 (UTC)

Thank you for this response. It is now clear to me what is the source of the disagreement - the term "double group" is ambiguous. Any direct product of two groups may be called a double group. However, I created this article to deal specifically with the application in magnetochemistry, which is the context of the chapter in Cotton's book. I propose the following resolution. 1) The article is renamed as "Double group (Magnetochemistry)". 2) The usage of the term is explained with reference to other meanings. 3) Restore the previous version of section 1, which contained a character table for D₄'. 4) Remove the two other sections.Petergans (talk) 09:35, 5 April 2022 (UTC)

I am not sure there is any consensus to Remove the two other sections. There's certainly an article on representation theory of SU(2) which is standard material. Robert Steinberg has given a uniform way to compute the characters table of any finite subgroup Γ of SU(2) (given its ADE labelling); there's also an article of Wulf Rossmann. Bertram Kostant has given a uniform branching rule for determining how irreducible representations of SU(2) restrict to Γ. Since the subgroups Γ and SU(2) deal with ordinary and spin representations, the latter are "double groups" or "double-valued representations" of Γ / {± I} and SO(3) = SU(2) / {± I}. Please make a report at WP:RSN is you have any questions about these 7 references, all of which are about double groups, aka double-valued or spin representations. Mathsci (talk) 20:54, 5 April 2022 (UTC)

Resolution

I don't disagree with the comments above, but the sections "Character tables of crystallographic point groups" and "McKay correspondence" belong in other places, not in this article. The text for applications has been restored and further revised, including a brief mention of the other meanings. The article has been renamed as "Double group (magnetochemistry)" to remove any ambiguity. Petergans (talk) 10:40, 11 April 2022 (UTC)

There has been no attempt to resolve things at all. The article is a sub-stub which reads like a blog post and not an entry in an encyclopedia There has been no attempt to read add any meaningful references in "group theory of physics", quite the opposite. The term "double group" was invented in 1929 by Hans Bethe, and arose naturally in physics and mathematics. Nobody would be able to tell that from this sub-stub. On the other hand the text book of F. Albert Cotton is a standard text book, referring to physics and mathematics, and natural scientists would have no difficulty reading that material. But that does not apply here, as there are no wikilinks to any other articles on wikipedia. The article has not improved since the AfD, so, as an intermediate step, I am proposing it for speedy deletion. Mathsci (talk) 11:55, 11 April 2022 (UTC)

This discussion is becoming heated, and I would argue for a compromise solution which distinguishes the 2 controversial sections. I would include "Character tables of crystallographic point groups" (and Hans Bethe) in the article, but exclude "McKay correspondence".

The reason for including crystallographic point groups is that their double groups have the same definition as the double groups of any molecular symmetry groups, namely the direct product of the symmetry group with the 2-member group whose elements are rotation by 2π and the identity operator. Crystallographic point groups are in fact a subset of molecular symmetry groups, corresponding to those symmetry groups which are compatible with the translational symmetry operations in crystals. This is a minor difference between the two concepts (molecular and crystallographic point groups) which belong in the same article, and we can mention that crystallographic point groups were the first molecular symmetry groups to be discovered, yes by Hans Bethe.

On the other hand the McKay correspondence has no obvious relation to the rest of the article so I think it does not belong in this article. It already has its own article which is sufficient. Dirac66 (talk) 01:04, 12 April 2022 (UTC)

Of course any point group {G} can be combined with {E,R} to create a double group, but I know of no application other than magnetochemistry. In table 10.6, p.252, Tsuckerblat gives details for Branching scheme of the double-valued representations of O'_h for sub-groups of O'_h. It appears to be purely of a predictive nature, so I suggest that the section as a whole does not belong in the WP article. Petergans (talk) 08:48, 12 April 2022 (UTC)

The presentation here seems like an impressionistic view of old lecture notes. With no attempt at clarity or being encyclopedic, Petergans continues to state that all references to physics are "irrelevant"; perhaps that might be his experience, but it is untrue on wikipedia (or academia). The "style" of editing adopted here is that of an engineering course (there were also unsuccessful attempts to import that style to group (mathematics)). The text book of Cotton does not adopt that approach and is well written. In contrast, Petergan's edits have no clearly formulated sentence on the representation theory of SU(2) or its Lie algebra, i.e. angular momentum and quantum mechanics, bread-and-butter material for second or third year undergraduate natural scientists. The claims of "irrelevance" continue here; at the moment the material is unverifiable from sources, and, without inline citations, seems just to be a rewrite of Petergans' old lecture notes. All mention of the words "group", "subgroup" or "representation" has been excised; Hans Bethe deserves better than that on wikipedia. After all, he was a physics Nobel laurate, so not somebody to be buried without trace. Mathsci (talk) 10:00, 15 April 2022 (UTC)

The binary subgroups Γ* ⊂ SU(2) are double groups that arise as finite subgroups Γ ⊂ SO(3). They are associated with regular polytopes (as explained in the classic book of H. S. M. Coxeter). For applications in physics or chemistry, there is a crystallographic restriction which dictates that the possible order of each group element is 1, 2, 3, 4 or 6.

Previously Petergans has written multiple times that the references to double groups appearing in Bethe's original article and subsequent text books are irrelevant: that appears to have been some kind of "culture gap" between physics and chemistry. Returning to the seminal 1929 article in Annalen der Physik (and its English translation), on page 16 Hans Bethe writes "We shall designate the group so obtained, which has twice as many elements as the original group, as the crystal double group, and shall inquire about its irreducible representations. This accomplishes our purpose of representing every element of the simple group by two matrices (a two-valued matrix). This procedure corresponds somewhat to the construction of the Riemann surface for the study of multiple-valued functions." On wikipedia, there are articles binary octahedral group, binary tetrahedral group and binary icosahedral group (all double groups). The subgroups ${\mathfrak {A}}_{4}$, ${\mathfrak {S}}_{4}$, ${\mathfrak {A}}_{5}$ ⊂ SO(3) have their corresponding binary subgroups ${\mathfrak {A}}_{4}^{*}$, ${\mathfrak {S}}_{4}^{*}$, ${\mathfrak {A}}_{5}^{*}$. Bethe discusses the quantum mechanical wave functions and tensor products of irreducible representations, citing Wigner & von Neumann (1928). F. Albert Cotton reproduces the character table of the icosahedral group, even though it is not a crystallographic point group (having 5-cycles (1,2,3,4,5) so elements of order 5 in ${\mathfrak {A}}_{5}$).

The representation theory and character tables of double/binary subgroups were explained in full detail by Francis D. Murnaghan in 1938, citing Bethe's 1929 paper. There is also a chapter on "crystallographic groups" explaining the crystallographic restriction (page 337). Tensor product rules are also described. Petergans has repeatedly claimed that this source is irrelevant; please see WP:CIR. Mathsci (talk) 19:45, 15 April 2022 (UTC)

Parenthetic comment. I've given lecture courses in Parts I, II and III of the Cambridge Mathematical Tripos over a long period, starting in 1990. The header = See also = breaks WP:MOS; the representation theory of SU(2) gives the correct character formula for $S^{m}(V)$, the $m$th symmetric power of the 2-dimensional vector representation $V$ evaluated on an a diagonal matrix with entries $e^{\pm i\theta }$; setting $j=m/2$ and $\theta =\alpha /2$ gives the usual quantum mechanical formula; and so on. On wiki.riteme.site, there is an incomplete mathematics BLP on me in which I have had no involvement; it lists some articles on mathscinet concerning projective unitary representations and character tables of binary finite subgroups. The article "double group" was listed for speedy deletion and then for deletion by D.Lazard. I mentioned my own knowledge of that topic at Wikipedia:Articles for deletion/Double group. The 1929 article of the physics Nobel laureate Hans Bethe was also mentioned; in "Applied Group Theory" by Arthur Cracknell, there is an English translation; later Bethe made his own independent translation. The topic has since been covered by numerous physicists using the term "double group" (cf references to books of Cracknell and Bradley & Cracknell); for example, here is a 1956 article by physics Nobel laureate C. N. Yang et al, "Strange Particles and the Conservation of Isotopic Spin", where the character tables of the "double groups" are treated, in particular for the binary tetrahedral group, binary octahedral group and binary icosahedral group. Other physicists like Wybourne and his collaborators have also worked out branching rules, tensor product rules, etc. In algebraic combinatorics/representation theory, mathematicians like Tonny Springer, Ian G. Macdonald, Bertram Kostant, Robert Steinberg, et al have explained systemically why the character tables of finite binary subgroups fit into an ADE pattern, including branching rules: that was first observed in 1969 by John H. Smith, who studied graphs of Perron-Frobenius eigenvalue 2 or less (cf Smith graphs); the character tables are the suitably normalised eigenvectors of the adjacency matrix of the Smith graph — the character ring of the subgroup has a basis consisting of irreducible characters and tensoring by the character of $V$ yields an adjacency matrix corresponding to an ADE graph. Mathsci (talk) 23:11, 18 April 2022 (UTC)

Content on "double group" by two physics Nobel laureates has already been mentioned here (Hans Bethe and C. N. Yang). The text books of Cracknell and Bradley & Cracknell are relevant; Cracknell's 1968 book is aimed at 3rd year undergraduates. Facts about the binary icosahedral group go back to an 1875 article of Felix Klein. The character table for the binary icosahedral group ${\mathfrak {A}}_{5}^{*}$ can be found in the paper of C. N. Yang et al, as well as this entry at nLab. In Cracknell's English translation of the 1930 article of Wigner, double groups are described in equivalent terms as double-valued or spin representations of the finite subgroups of SO(3). In Wigner's book on group theory, later translated into English, double group is explained in terms of the double cover SU(2) of SO(3); in undergraduate courses, that double cover is often described in terms of unit quaternions or the adjoint representation of SU(2). Mathsci (talk) 10:43, 21 April 2022 (UTC)

Expert tag

This tag has been removed as I am an expert in the applications of group theory in science, e.g. my book "Vibrating Molecules", (1971) chapter 3, "Molecular Geometry", pp. 53-77. Petergans (talk) 08:27, 22 April 2022 (UTC)

As explained on User:Hammersoft, the tagging is for an expert in physics (or mathematics). The current writing on this stub indicates WP:CIR. The repeated claims to be an expert on group (mathematics) do not seem to be supported by edits; so far, no coherent sentence has been produced about the representation theory of SU(2), even in the lede. The book of F. Albert Cotton is specifically mentioned by Cracknell as being the few text books written to be accessible to undergraduates in physical chemistry. Like Eugene Wigner's book, Cotton's book is available and it's easy to check what's written there, especially about Hans Bethe. So far, it does not tally with what has been written in the article. If another reference has been used for the article instead of the WP:RSs, that would require further discussion. The usual place to check is WP:RSN or in a book review. Mathsci (talk) 09:27, 22 April 2022 (UTC)

You keep raising CIR: I'm seeing a nontrivial question of content and a behavioural issue from two editors who are normally competent, but do not get along, and so are failing to resolve a tricky but fundamentally resolvable content issue. The way you characertise the dispute suggests a mindset that the kind of issue on display here must come down to a determinate failure of good will or competence on some party to the dispute: if this is along the right lines, I think this is a perfectionist way of looking at the matter that isn't helpful to dealing with the messy nature of human interaction. While I think you make good points about the need for content to respect the unity of knowledge, I think you are disrespectful of Peter and unduly dismissive of the idea that presenting the applications in a way that is accessible to the readership most likely to be interested in this content is difficult to do with the approach you prefer.

I hope we don't need another expert since we are unlikely to get someone with more relevant expertise than the two of you have. Instead either or both of you adjusting your own behaviours is a thing that, while not easy, does not demand wishing for ponies: I think you need to be more accommodating, while I think Peter's vision of how to deal with the conflict has "ownership" issues. There might be tricks we can use to help, the bureaucratic device of RfCs seems to be in vogue at the moment, but there's a risk of them being a complete waste of time if the overall level of respectfulness is not improved. — Charles Stewart (talk) 11:08, 22 April 2022 (UTC)

Normally on wikipedia, I avoid editing content directly related to my research; on this very rare occasion, as mentioned on Hammersoft's talk page, my own knowledge/expertise has come into play, as stated in the AfD. Because I don't wish to discuss this on wikipedia, I have used the usual method of summarising WP:RSs that can be verified. Mathematically the character table of the binary icosahedral group corresponds to that of SL₂(F₅), the binary octahedral subgroup to that of SU₂(F₉) and the binary tetrahedral group to that of SL₂(F₃): all of these were calculated by Frobenius in 1899; and later independently by Schur and H.E. Jordan in 1907. Later presentations have used the oscillator representation (for finite fields) of André Weil. The character table of the double group of the icosahedral group appeared in "Stability of Polyatomic Molecules in Degenerate Electronic States II —Spin Degeneracy", H. A. Jahn, Proc. Roy. Soc. London (1938), pages 117–131. This has later been applied in Material Sciences to fullerene; see J. R. Heath, S. C. O’Brien, Q. Zhang, Y. Liu, R. F. Curl, H. M. Kroto, F. K. Tittel, and R. E. Smalley (1985). J. Am. Chem. Soc. 107, 7779 (three chemistry Nobel laureates). The character tables of the double group are given in K. Balasubramanian (1996). Chem. Phys. Lett. 260, 47, along with many, many other mathematics/physics books or articles. See also, "Case, K. M.; Karplus, Robert; Yang, C. N., "Strange particles and the conservation of isotopic spin", Phys. Rev. 101 (1956), 874–876".[2] Whatever the interpretation, this is fin du siècle mathematics: it has been forgotten or misremembered many times, only to be reinvented in different guises; nobody can claim to be an expert. Mathsci (talk) 04:56, 23 April 2022 (UTC)

Fullerene complex

Following from the comment above, Greenwood & Earnshaw, 2nd, p 288, state that lanthanide ions that are encapsulated in the C60 fullerene should be designated as [La³⁺@C₆₀³⁺]. The Ce³⁺ complex will have the f¹ configuration that is mentioned in the article and so that compound should to be treated using the icosahedral double group. I will add this example, together with reference to Balasubramanian, when I have found a specific reference for the cerium(III) complex. Petergans (talk) 09:07, 23 April 2022 (UTC)

@Petergans: there has been no acknowledgement that you have got this very wrong. Above I have written careful statements about the character tables of the binary icosahedral group SL₂(F₅) and the other two 'exceptional' binary subgroups. The title will now almost certainly have to be adjusted, because magnetochemistry is not nanotechnology. That material has also been presented by Harvard/MIT academics like Shlomo Sternberg and Bertram Kostant. I have also purchased the short 1971 book, "Vibrating Molecules" for £2.51. Like other chemistry text books, it does not give an account of the character table of the binary icosahedral group; that applies in particular to the text book of Greenwood & Earnshaw, where there is no discussion of character tables, double groups, etc. The article of C. N. Yang et al does give the tables; but of course they can be found in the original references from 1899 and 1907 of Frobenius, Schur and Jordan. They introduced and developed that theory and it has not been superseded. Mathsci (talk) 10:48, 23 April 2022 (UTC)

The character tables of the finite double subgroups, in particular the double icosahedral group, are known to be highly significant scientifically; that became common knowledge in the 80s and is clearly presented in various articles on wikipedia. Independently the character theory of non-crystallographic point groups become relevant in physics (Yang et al) and chemistry (Kroto, Curl and Smalley), as mentioned above. The particular edit[3] was reverted because it was unsourced WP:SYNTH and WP:OR. I have the book of Greenwood & Earnshaw available as an OCR pdf file, so could verify that neither double groups nor character tables were mentioned. The sourced content I wrote here was carefully prepared over several hours; if it was written by me, so why suggest otherwise? The edit here[4] to a user page would normally be reverted by the user or an administrator. The edit reads: [[User:mathsci]] has removed the contents of my last edit. The article cited in the deleted text was entitled "Double group of the icosahedral group (I_h) and its application to fullerenes". Intolerable! There seems to be a "preferred version" User:Petergans/sandbox, which looks like WP:OWN.

I already wrote, "I avoid editing content directly related to my research; on this very rare occasion, as mentioned on Hammersoft's talk page, my own knowledge/expertise has come into play, as stated in the AfD." In real life, my own research partially involved the character tables of the binary subgroups, with two articles in Ann. Math. and Invent. Math., resulting in a prize. Mathsci (talk) 12:57, 24 April 2022 (UTC)

• As a general comment, on google scholar or mathscinet (with "anywhere" in different entries), there are many references for "double icosahedral group" which simply list the character table. The character table of SL₂(F₅), a finite group of Lie type or Chevalley group, has been known since Schur and H. Jordan (1907). The two other binary exceptional are of Lie type, but the subgroups (modulo their centre) are solvable, unlike PSL₂(F₅), which is simple. This is well-trodden territory and there are plenty of expositions which are not ad hoc for SL₂(F_p): in English, there is H. Jordan, 1907; or Dornhoff's English version of Schur, 1907; or Tonny Springer's exposition in "characters of special groups"; or lecture notes of Prasad and Jeffrey Adams; or using the method of J. A. Green for GL₂(F_p) and the reduction trick to SL₂(F_p), first proved by Karkar; or using the methods of Andrey Zelevinski; or the expositions of Piatetski-Shapiro/Paul Garrett for GL₂ and SL₂; an elementary account by Luisa Aburto-Hageman; the American Mathematical Monthly article by J. E. Humphreys [5]; or N. Backhouse and P. Gard.[6] In these accounts, the Steinberg representation (1951–1957) appears, as well as the whole machinery of Deligne, Lusztig, Drinfeld, etc, explained in the books of Carter, Geck & Malle, etc. Transversally, Tonny Springer's gave a uniform treatment of character tables for 'exceptional' binary subgroups.[7] Mathsci (talk) 00:08, 25 April 2022 (UTC)
• Fulton & Harris, "Representation Theory: A First Course" also covers GL₂(F_p) and SL₂(F_p) very well. "Tensor products" intervene through the usual physics interpretation of correlation coefficients (special functions or the wave functions of quantum mechanics, as in Weyl, Wigner, Bethe); see also Hamermesh's "Group theory and its applications to physical problems" (used by Balasubramanian[8]). An elementary derivation of tensor product rules was computed by Luisa Aburta-Hageman and Jose Pantoja.[9] These can also be found in the uniform treatment of Tonny Springer[10] who determined the restrictions to the (exceptional) binary subgroups of the m-dimensional irreducible representation of SU(2); induced representations of characters of the diagonal group and Frobenius reciprocity imply tensor product rules. See also the related articles of Steinberg[11] and Kostant.[12][13][14] See also Happel, Ringel et al.[15] For double icosahedral group and buckeyballs, see the article of Fan Chung, Kostant and Sternberg.[16][17] Mathsci (talk) 19:00, 25 April 2022 (UTC)

Thank you for these citations. The background that is covered by the cited articles is certainly of interest to students of mathematics, but is all but unintelligible to others. For example, I don't know what GL₂(F_p) and SL₂(F_p) mean.

The issue, as I see it, concerns the extent to which mathematical details belong in this article. It's a very specialized topic. For example, S.F.A. Kettle's book "Symmetry and Structure, readable group theory for chemists" 2nd ed. Wiley, 1995, contains much mathematical detail, but makes no mention of double groups.

In general terms, applications of group theory are relevant in both physics and chemistry, so the "physics" tag is irrelevant and will be removed. Petergans (talk) 10:26, 26 April 2022 (UTC)

• At this stage there are issues of WP:CIR with the edits of Petergans. Facts on wikipedia about finite groups are easy to find; there is no need for a particular editor to present themselves as some kind of expert, since that might or might not be the case. The representation theory of finite subgroups of SU(2) was determined by Schur and Jordan; the realisation as linear groups over a finite fields can be read in the classic "Theory of Finite Groups and Applications" (1916) by G. A. Miller, H. F. Blichfeldt and L. E. Dickson — it is well known, encyclopedic and easy to find on wikipedia. It is also easy to find in the literature, provided that wikipedians look for good reliable sources. The 1961 CUP book "The Theory of Transition-Metal Ions" by the chemist John Stanley Griffith (1928–1972) is a good source by an expert; it gives the character tables of binary subgroups and their tensor product formulas (Griffith uses the notation K and K* in Appendix A.7, A.8 and A.9); in 1993 the chemists Fowler and Ceulemans generalised the formulas in Theor. Chim. Acta. Mathematical and theoretical physicists have also revisited and generalised that theory. Kostant and Sternberg were amongst those. Pierre Ramond's 2010 CUP book "Group Theory: A Physicist's Survey" contains a concise and easily readable account of that with character tables and tensor product formulas: the most general rules can be found in Cummins & Patera, Luhn & Ramond and Everett & Stuart. Griffith was a fellow of King's College, Cambridge, worked with Francis Crick, but then became a professor at Indiana University; he died at the age of 44. A layman's account of Buckminsterfullerene and the binary icosahedral group is described in John Polkinghorne's 2008 OUP book, "Meaning in Mathematics": it's written by Mark Steiner, who refers to Chung, Kostant and Sternberg. The chemist Krishnan Balasubramanian seems to have rediscovered the representation theory of the binary icosahedral group in 1996; there is very brief review of that in this 1997 book, "Relativistic Effects in Chemistry, Part A"; Griffith's 1961 account seems more encyclopedic. Mathsci (talk) 08:22, 27 April 2022 (UTC)
• Puzzlement. Petergans writes that they do not know what GL₂(F_p) and SL₂(F_p) mean. Modulo p is often taught at UK sixth forms for maths A levels. The 2005 book "Group Theory with Applications in Physical Chemistry" by chemist Paul Jacobs treats double groups, and even gives the character table of the binary icosahedral group, denoted by Y (A3.9, page 465); modulo p is taken for granted. It's quite hard to reconcile having an in-depth knowledge of group theory, including induced representations, while having no familiarity at all with Z modulo p, i.e. the finite field F_p. Mathsci (talk) 13:49, 27 April 2022 (UTC)

He didn't say he doesn't know what modulo means. He asked what GL₂(F_p) and SL₂(F_p) mean. It is not obvious that these symbols have anything to do with a modulo. If you want to insert these symbols into an article, then you should explain them. Dirac66 (talk) 14:09, 27 April 2022 (UTC)

The articles general linear group, special linear group, projective linear group and field are all easy to find, and are not "specialized". Similarly the term group representation and homomorphism; and projective representation. In Paul Jacobs' book "Group Theory with Applications in Chemical Physics", this terminology is used, often as chapter headings, and is designed to "appeal to advanced undergraduates and graduate students in the physical sciences". As stated there: "Group theory is widely used in many branches of physics and chemistry, and today it may be considered as an essential component in the training of both chemists and physicists. This book provides a thorough, self-contained introduction to the fundamentals of group theory and its applications in chemistry and molecular and solid state physics." John Stanley Griffith is both a mathematician and a theoretical chemist. Pierre Ramond is a distinguished living theoretical physicist; his book, "Group Theory: A Physicist's Survey", has chapter headings "PSL(7)" and "Matrices over Galois fields"; it gives an up-to-date treatment of the representation theory of the finite subgroups of SU(2). I am waiting to read Peter Gans' book "Vibrating Molecules", priced at £2.51. There is no obfuscation in Griffith's 1961 account; in a footnote he writes: "The first and most important general discussion of the application of the theory of finite groups and their representations to the theory of ions in crystals, was given by Bethe, Ann. Phys., Lpz. (1929),3, 133. The concept of a spinor group is due to Cartan. The characters of the irreducible representations of the finite spinor groups T*, 0* and K* were first given by Frobenius, S.B. preuss. Akad. Wiss. (1899a), p. 339." Mathsci (talk) 18:58, 27 April 2022 (UTC)

---

Observations. I have consulted the book of Peter Gans, purchased for £2.51 from an online bookseller in Lincoln. Like the stub, it is poorly written mathematically, with no proper definitions of groups, representation theory, etc. By comparison the 1961 book of John Stanley Griffith has clear definitions, with no obfuscation. Griffith was a trained mathematician (a wrangler from Trinity College, Cambridge), who went on to work in chemistry, physics and biology (Crick & Watson). That explains why he writes so well, and why his account of double groups is reliable: he gives a complete account of the non-crystallographic double icosahedral group, including its character tables and tensor/Kronecker product rules. As said before, these can also be found elsewhere, e.g. in Pierre Ramond's book or Paul Jacobs' book. In contrast, I can find almost nothing useful about double groups in the book of Peter Gans. There is no mention of the double icosahedral group; the double octahedral group is given as a direct product, so does not match up with Griffith's character tables, the character table for nLab here or standard references in mathscinet. Page 217 of Gans' book has a 10 x 10 table (the character table of the direct product of the octahedral group and Z mod 2), while the binary octahedral group has an 8 x 8 table with degrees 1, 1, 2, 3, 3 (ordinary reps) and 2, 2, 4 (spin reps).

I noticed that in a previous discussions from 2013, there were disputes where Petergans argued about the article Savitzky-Golay filter and that article talk page; there were unsuccessful attempts to create forked content – Numerical smoothing and differentiation. There was similar disruption on Convolution, where Petergans decided to contradict the definitions of mathematicians, in particular standard references of Laurent Schwartz and Lars Hörmander (two Fields medallists). In Talk:Convolution#Unacceptable reversion, the edits reached an impasse.[18] On that talk page, statements were made that were mathematically incorrect (the correct framework is given in many places, e.g. "Abstract Harmonic Analysis" by Lynn Loomis or Walter Rudin's "Fourier Analysis on Groups").

Double groups are dealt with very well by others, but not in the book by Peter Gans: it fails WP:RS from the point of view of mathematics or physics. There is a review in the Journal of Molecular Structure; the reviewer writes:

"The aim of this book is to give undergraduate and postgraduate students an introduction to vibrational spectroscopy in inorganic chemistry. Therefore, although the theories treated in this book are general, their applications are limited to inorganic compounds. The problems of vibration-rotation are also omitted. No previous knowledge of the subject being assumed, the first part of the book is devoted to an explanation of the mathematics used in molecular vibrational problems. This is followed by an account of group theory and its application to molecular vibrations as well as normal coordinate analysis; the text is illustrated by reference to simple molecules [...] The problem of crystal spectra is beyond the scope of this book. However, since most inorganic compounds are crystalline at room temperature, some reference to this topic would have been helpfu1."

Mathsci (talk) 11:40, 4 May 2022 (UTC)

General revision

The version 12 May 2022 contains deficiencies, including

• Incorrect chemistry in the abstract: "complexes of metal ions that have a single unpaired electron in .."
• Removal of valid icosahedral example, image and relevant citations
• Re-introduction of gold(II) without there being any relevant example

The presentation of character tables and multiple mathematical symbols is badly degraded.

These deficiencies have been corrected and the whole article has been revised for greater clarity of presentation. Petergans (talk) 09:46, 14 May 2022 (UTC)

---

The character tables of double tetrahedral, octahedral and icosahedral groups following Frobenius (1899)

Petergans's edits fail to present mathematical material about the character table of the binary (or double) icosahedral group, first proved in 1899 by Georg Frobenius in terms which have not changed since present days. Online sources are here in the "Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin", "Deutsche Akademie der Wissenschaften zu Berlin", 6 April 1899, pages 330–339. The character tables of the binary octahedral, tetrahedral and icosahedral groups are given on Page 339.[19] The 1961 chemistry WP:RS of John Stanley Griffith gives a full mathematical account, including the character tables. The "presentation" in User:Petergans/sandbox is abysmal: no explanation has been been given about why a non-crystallographic point group occurs. Griffith's 1961 book gives a self-contained account of double groups, which cites Frobenius, Wigner, Bethe, etc. The sandbox shows a failure or refusal to use WP:RSs; the content itself fails [[WP:V]. On the other hand, the book of Griffith is excellent and readable for general scientists. The application to buckminsterfullerene has already mentioned on this talk page, with references, but the sandbox gives no proper explanation. Carbon 60 has been discussed in an article by the Nobel laureate C. N. Yang referring in particular to character tables; it is unclear whether this applies to the icosahedral group (and its direct product by the cyclic 2-group) or to the binary icosahedral group. I found an insubstantial note of Krishnan Balasubramian from 1996, but noted it was not a reliable source, with incomplete or inaccurate results about characters and their product rules. Griffith, Ramond, Jacobs and others have given clear accounts of double groups; the suggestion that there might be a completely separate theory of double groups seems like something from the Alice books of Lewis Carroll. Mathsci (talk) 12:05, 14 May 2022 (UTC)

Sandbox

Note that the preferred version of Petergans can be found here in their sandbox. Mathsci (talk) 11:47, 4 May 2022 (UTC)

The most recent version is here.[20] Mathsci (talk) 20:07, 8 May 2022 (UTC)

Zirconium with 4f electrons???

The last sentence in the article now says: "Complexes of zirconium (III), with 13 electrons in the 4f valence shell may also be treated using a double group." Zirconium of course has no 4f electrons. Should this perhaps read hafnium, which is in the same column of the periodic table? Could someone with a copy of Greenwood and Earnshaw check the source? Dirac66 (talk) 19:24, 15 May 2022 (UTC)

I have done so. The sentence has been deleted.Petergans (talk) 09:20, 19 May 2022 (UTC)

Crystallographic restriction theorem

The icosahedral group is not a crystallographic point group, because it contains elements of order 5 – in the crystallographic case, only elements of order 2, 3, 4 and 6 are permitted. In the 1938 book of Francis Murnaghan on "The Theory of Group Representations", Chapter XI is entitled "The Crystallographic Groups". That is one of the many sources where that material is explained. Facts about crystallographic point groups are fairly well known and not subject to debate. The 32 point groups are tabulated in the 1963 WP:RS of Koster, Dimmock, Wheeler and Statz. Mathsci (talk) 18:07, 19 May 2022 (UTC)

What has this got to do with double groups? Petergans (talk) 21:03, 20 May 2022 (UTC)

For crystallographic point groups, see Page 80 of your own 1972 book. Double groups are subsumed in Frobenius' work on character tables of the finite subgroups of SU(2), the binary groups — so cyclic groups, binary dihedral groups, and the exceptional three binary subgroups (tetrahedral, octahedral and icosahedral). The character table also includes the table for the finite subgroups of SO(3), which give a trivial double group corresponding to the product of the subgroup and the cyclic 2 group. The crystallographic restriction means that only certain finite subgroups of SO(3) have to be considered, as reproduced in multiple sources (in your book you list the 32 cases). So the theory of double groups in this context was already established in 1899. That was explained in Cotton's book who refers back to Griffith's 1961 book. The case of quasiperiodic crystals or quasicrystals involves the icosahedral group and, according to Ted Janssen, the character table of its double. Mathsci (talk) 22:10, 20 May 2022 (UTC)

You make the case for me! Point and space groups are fully dealt with in other WP articles. Including them here serves no purpose; indeed, it is a potential cause for confusion. At most, they require a "see also" entry. Please note that the suggestion by Janssen is purely theoretical. Crystalline icosahedrite has the space group 5 3m , which is not a double group. Petergans (talk) 07:26, 21 May 2022 (UTC)

Thank you for your input here. Each wikipedia article is normally self-standing, since wikipedia is not a source; WP:RSs often have to be provided for each particular topic or wikilink. For example for crystallographic restriction theorem, wikipedians have provided their own ad hoc mathematical explanations, which sometimes match up with sources; the methods of H. S. M. Coxeter from the early 1930s, however, adopt a by-now classic approach (cf Murnaghan); possibly they have been rediscovered a decade later by crystallographers. Providing a context for binary or double subgroups – with its character tables, branching rules and tensor product rules – enables the article to become encyclopedic.

The footnote on quasiperiodic crystals and quasicrystals goes in a different direction—material sciences. The theory goes back to Roger Penrose and quasiperiodic tilings ("pentaplexity") and has been described in many places, e.g. solid state physics: "Exactly Solvable Models" and the Yang-Baxter equations, with the 2013 survey, "Quasicrystals – The Impact of N. G. de Bruijn", by Helen Au-Yang and Jacques Perk. Ted Janssen died in 2017, so it's hard to comment in these circumstances. Mathsci (talk) 12:06, 21 May 2022 (UTC)

Consensus impossible

After many of my recent edits has been reverted, I conclude that it is impossible to reach consensus. In consequence, I will not make any further attempts to improve this article. Petergans (talk) 17:44, 23 May 2022 (UTC)

You deleted all references to Hans Bethe: how was that an improvement? It's easy to give a short description of the representation theory of SU(2) including facts about Lie algebras, normally described using the angular momentum operator in quantum mechanics. Mostly you have reverted the tag requesting the content concerning SU(2) and its characters be improved. For finite subgroups of SU(2) (or SO(3)), character tables make no sense infinitesimally. Instead, it is true that every group element of SU(2) is conjugate to a diagonal element; and there are analogous statements for conjugacy classes of finite subgroups. The main point is that that material exists; but it has been systematically deleted as "irrelevant". The sandbox has only been added to; there have been no deletions, except for the 1996 reference to Krishnan Balasubramanian, which did not match up to more reliable sources. Mathsci (talk) 21:36, 23 May 2022 (UTC)

sandbox/notes + removal of references

At the moment User:Petergans is substituting User:Petergans/sandbox/notes for this article. Where there are references, they have been deleted, instead adding "citation needed", even when three text books cover that material. In the sandbox/notes, there appears to be some attempt to summarise the 19th-century mathematics of Klein and Frobenius. However, as is well known, the subject began with the work on binary invariants due to Clebsch and Gordan. Frobenius derived and tabulated the character tables of all finite subgroups of SU(2): that is currently reported on wikipedia, without suggesting the opposite. Similarly facts about platonic solids can be found directly on wikipedia and go back to Plato and Ancient Greece The 1875/6 Math. Ann. book is subsumed by Klein's 1884 book on the icosahedron and in particular describes the finite subgroups of SU(2). The binary subgroups have their own article, e.g. binary icosahedral group. The statement "The magnetic properties of this ion are treated using the icosahedral double group" quoted from "Fullerenes with metals inside" has no reference at all to the icosahedral group as an OCR search shows. Creating that sentence was not helpful.

In 2013, there were similar problems with the article "convolution"; there was attempted forked content, which had to be deleted.

The refusal to use standard vocabulary and wikilinks, adopted on wikipedia, is part of the problem. As far as representation theory is concerned, the accounts following Klein, Frobenius and wikipedia are correct; alternative unsourced versions are unreliable as they fail WP:RS and WP:V. If this subject was invented by 19th-century mathematicians, there is no reason to suggest otherwise. Mathsci (talk) 19:29, 1 June 2022 (UTC)

Point group I'h

The character table for the point group labelled I'_h was obviously wrong: one column was labelled 1C₄^[2] (not 15C₄...) and there should have been 10 columns of characters, not 9. In any case this was the table for the point group (in modern notation) I_h, which is not a double group in the modern sense. This character table and the character table for the group I (incorrectly labelled I_h) have been deleted. In this article a double group is one in which rotation by (360+x)° and identity belong in a separate classes, as illustrated in the table for D'₄, below.

The tensor product rules are irrelevant in the context of this article. Petergans (talk) 16:24, 10 June 2022 (UTC)

See WP:CIR. In the binary icosahedral group, the element –I is central. The character table of the binary icosahedral group, isomorphic to SL₂(F₅) or A₅*, is given in numerous accounts, including in the article itself, with the page written by Frobenius. Mathsci (talk) 17:32, 10 June 2022 (UTC)

@D.Lazard: please feel free to comment about the character table of the binary icosahedral group. So far you have written nothing at all. Deleting the character table with all the citations was disruptive vandalism. My Inv. Math. paper reproduces the character table. Similarly the 1956 of C. N. Yang. And so on. Mathsci (talk) 17:23, 10 June 2022 (UTC)

The binary icosahedral group has exactly nine distinct irreducible representations and exactly nine conjugacy classes. Mathsci (talk) 17:48, 10 June 2022 (UTC)

To editor Mathsci: I have read WP:CIR; in section WP:NOTCIR of this article, one reads Calling someone incompetent is a personal attack and is not helpful. Always refer to the contributions and not the contributor, and find ways to phrase things that do not put people on the defensive or attack their character or person. Apparently you ask people to read articles that you have not read or that you do not understand. Please, stop accusing of incompetence everybody who disagrees with your edits. This is WP:personal attacks that are forbidden here. Also stop your boasts about your mathematical competences (talk at ICM, Inv. Math. paper, etc.); they do not matter here, and cannot be verified, as Mathsci is not your real name. In any case, even if you were a good mathematician, this does not prove that this is still the case. You have been blocked many times and IBANed for behaviours similar to your behaviour here; each time you promised to stop such behaviors. Apparently, you have not the competence for respecting your promises. D.Lazard (talk) 18:34, 10 June 2022 (UTC)

Previously on this article talk page, it was stated: there is a 1956 article by physics Nobel laureate C. N. Yang et al, "Strange Particles and the Conservation of Isotopic Spin", where the character tables of the "double groups" are treated, in particular for the binary tetrahedral group, binary octahedral group and binary icosahedral group. The article is linked here, the tables are clearly displayed and marked as "double icosahedral group". Mathsci (talk) 18:54, 10 June 2022 (UTC)

In the character table for I_h in Cotton, Appendix 3, there are 10 classes. The character table is concordant with the one in the old article, apart from the missing column there. The same table in Cotton shows the charcters for point group I as a sub-group of I_h.Petergans (talk) 20:13, 10 June 2022 (UTC)

That is because there are two double groups for each subgroup Γ of SO(3) as currently explained in the article. Cotton refers for details to Griffith's 1961 book, calling it a "very sophisticated treatise". The two double groups are: (1) the trivial double group Γ x { ± I } (with character table given by the product of the character table of Γ and the cyclic 2 group); and (2) the subgroup Γ' of SU(2) covering Γ ⊂ SO(3) twice. They are not the same; and C. N. Yang et al accurately described the character table of Γ'. In Griffith's treatise, he writes K for the icosahedral groups and K* for the double or binary icosahedral group (i.e. Γ'). Mathsci (talk) 22:16, 10 June 2022 (UTC)

Lead

I see that Mathsci has greatly improved the lead. Nevertheless, the first paragraph remains too WP:TECHNICAL, and the second and third paragraphs do not belong to the lead and should be moved in a history section. Moreover the link between the mathematical theory and the physical applications is insufficiently explained.

Independently, I have written an other version of the lead in User:D.Lazard/Double group. It certainly deserves to be improved, but, IMO, it is what readers need for the lead of this article. Your advices are welcome. D.Lazard (talk) 15:02, 15 June 2022 (UTC)

The first paragraph of the lede is actually my fault. I have no strong attachment to it; the text was merely my first, off-the-cuff attempt at an introduction that would benefit somebody, since I couldn't tell what the purpose or intended audience of the previous lede was. I tried to cook up an opening sentence at the Scientific American or Quanta Magazine level and follow with some text that would be comprehensible to students who have seen a little group theory. I think Wikipedia house style is to put the bold bit as early as possible, rather than building up to it (which is more essay-like), but as I said, I've no strong feelings on the matter.

I agree that the second and third paragraphs belong in a "History" section (actually, the coverage of the history, the mathematics, and the applications are rather jumbled throughout the page). XOR'easter (talk) 17:59, 15 June 2022 (UTC)

OK, for the fun of it, I blended your suggested lede into the current one and tried my hand at reorganizing the material. There are still passages that strike me as needlessly opaque; for example, discussing all the different ways that the representation theory of SU(2) can be developed seems beside the point in this article. Indeed, the jump from finite-order subgroups to finite-dimensional representations of Lie groups is rather jarring. But one thing at a time. XOR'easter (talk) 18:41, 15 June 2022 (UTC)

Another thing that is currently rather unclear is which subfields actually use the "double group" terminology. I don't think I'd ever seen a paper about or involving a binary polyhedral group that actually used the specific term "double group" until I started working on this article. Spinors, yes; double covers, yes; but not double groups. The contents of this article should be focused on those aspects of the mathematics employed by the people who use the terminology. XOR'easter (talk) 21:01, 15 June 2022 (UTC)

You are absolutely right. I created this article to fill a very small gap, using mainly Cotton's book as the main source of information. I also Used Tsukerblat's book. After some additions by user:mathsci, which appeared to be irrelevant, I changed the name to "Double Group (Magnetochemistry)". After about one month the name was changed back by mathsci, who then added more of the the mathematical stuff. You are correct: that material is outwith the original context of the article. There is no mention of any of it in Cotton or Tsukerblat. It belongs in group theory or a sub-article thereof. Mathsci has reverted very many of my edits. So, with great reluctance, I support the proposal that mathsci be banned from editing this article. If that occurs, I would reinstate the version in my sandbox, after some minor clean-up. N.b. previously, mathsci had added two obviously irrelevant sections and has not contested their deletion. Petergans (talk) 08:10, 19 June 2022 (UTC)

Mathsci is banned from English Wikipedia for an indefinite period of time. So, he will not edit here further.

IMHO, your sandbox version does not respect the manual of style for the lead (MOS:LEAD). In particular, it is too technical, and the context is insufficiently explained. I suggest you to keep, and possibly modify the present lead, instead of restoring the lead of your sandbox D.Lazard (talk) 09:02, 19 June 2022 (UTC)

Thank you. I will work on my sandbox to try to meet the lead criteria and other content. Petergans (talk) 09:19, 19 June 2022 (UTC)

Fullerines

The encapsulated cerium(III) complex described in the article has a single 4f electron in the cerium atom and at least one unpaired electron in C₆₀^3-. It is the fact of these that properties that dictates that a double group is needed. All fullerenes without an encapsulated metal ion are diamagnetic and therefore irrelevant in this article. Their symmetry is that of a point group. Petergans (talk) 08:49, 16 June 2022 (UTC)

The lines about fullerenes and quasicrystals are hard to follow and seem to oversell the historical significance of the mathematics over the chemical synthesis. XOR'easter (talk) 22:51, 16 June 2022 (UTC)