User talk:JackSchmidt: Difference between revisions

Content deleted Content added

Inline

Revision as of 23:32, 20 January 2008

Welcome

Welcome!

Hello, JackSchmidt, and welcome to Wikipedia! Thank you for your contributions. I hope you like the place and decide to stay. Here are some pages that you might find helpful:

I hope you enjoy editing here and being a Wikipedian! Please sign your messages on discussion pages using four tildes (~~~~); this will automatically insert your username and the date. If you need help, check out Wikipedia:Questions, ask me on my talk page, or ask your question and then place {{helpme}} before the question on your talk page. Again, welcome! WLU 23:14, 3 November 2007 (UTC)[reply]

List of snowclones

Hi Jack

I'm really sorry! I missed your first comment on my talk page. It's been quite a busy few days on there and as it wasn't at the bottom, I didn't spoit it. I have userfied the content of the list to User:JackSchmidt/List of snowclones. Please do not restore it to the main article space, or add huge swathes of it to the snowclones article. Thanks. Neil ☎ 17:38, 4 November 2007 (UTC)[reply]

Thanks very much; it looks great! No worries about the missed comment or about an attack of the snowclones. My wife and I think they are great, but I think a giant list is better for a fan site than an encyclopedia. JackSchmidt 18:26, 4 November 2007 (UTC)[reply]

Agreed. And anyone who makes a Star Wars pun is alright with me! Neil ☎ 21:05, 4 November 2007 (UTC)[reply]

Math edits (Nbarth)

Hi Jack—glad you enjoyed the contributions, and thanks for the kind words. I'd just wanted to understand Clifford algebras, and way lead to way. (Ok, numerical polynomials are not part of that—I just remember Peter May talking about them. But I hope the article grows!)

I've thought a lot more about dicyclic groups and understand the connection to pin groups much better, so I'll shortly (tomorrow, I plan) write up the connections (and the center of the pin groups).

Thanks again for the support! Nbarth (talk) 01:34, 25 November 2007 (UTC)[reply]

Franklin

Hi, I think you need to see read the paper. http://www.jstor.org/view/00029890/di991539/99p0082c/0 —Preceding unsigned comment added by Franklin.vp (talk • contribs) 22:03, 29 November 2007 (UTC)[reply]

The paper above reference some early papers. I saw only one of them (reference number 5) not precisely related to this theme. But reference number 2 (L. E. Dickson Linear Groups ) it is acknowledged to be a very good book. I haven't seen it. I don't what is the approach there.

Exceptional outer automorphisms

Hi Jack,

Re: exceptional outer automorphism (& your note to me). I think I added both that line and the {{Fact}} tag, as I had heard or read that statement, but didn't know a reference. It's an interesting statement, hence I wanted to include it, but it's a strong statement, hence I wanted a reference.

Your additions (reference, connection to PSL(2,9), and caveat about sporadic groups) are all excellent and exactly what I was hoping for. I've incorporated them into the running text—hope you like it.

The connection with Out(PSL(2,9)) is very nice—it can be interpreted as "the outer automorphism is no more or less exceptional than the isomorphism with PSL(2,9)".

…and good point about sporadic groups—how many outer automorphisms "should" a Mathieu group have?

Thanks again!

Nbarth 16:29, 2 December 2007 (UTC)[reply]

Thanks for support re: Categories of manifolds

Hi Jack,

Just noted your kind note re: the article "Categories of manifolds"—thank you. As you say, many of the things I've found most valuable (and enjoyable) in math are these syntheses, these overviews or connections, and many hard-nosed people look down on these as woolly, both in math and on Wikipedia.

(One synthesis/idea that I very much like is the "Field with one element". I was also delighted when I observed that index p normal subgroups form a projective space, which is a hint/indication of the connections of a general finite group to Lie theory/Lie type groups.)

I plan to rework the article and include the more rigorous parts in appropriate places, but retain the broad sweep either on a user page or on my personal home page.

…and yes, as I've gone along, I've often found that not only was some connection known classically (but has been forgotten in a more axiomatic age), but was in fact the original point of view or motivation! (Which, if nothing else, makes my work more scholarly.)

Nbarth 16:42, 2 December 2007 (UTC)[reply]

Magma

Thanks for finding that bit of the Magma manual for me. Obviously I was very naive to assume FreeLieAlgebra would return a Lie algebra! :) 163.1.148.158 (talk) 12:33, 11 December 2007 (UTC)[reply]

p-group article

Could you take a look at this edit? I started to try to copyedit it and incorporate it into the article, but after taking a closer look I was tempted to revert it. If I understand it correctly, it attempts to show in a specific case that having prime power order implies being a primary group. But this should be an immediate consequence of Lagrange's theorem. I would think the converse, the claim that finite primary groups have prime power order, is slightly more interesting. Michael Slone (talk) 14:56, 11 December 2007 (UTC)[reply]

I think the edit is definitely good faith, and definitely requires copyedit. However, I agree the earlier statements "If G is finite, this is equivalent to requiring that the order of G (the number of its elements) itself be a power of p. ... the dihedral group Dih4 of order 8 is a non-abelian 2-group." subsume this edit entirely, other than confusing statements about groups of order pq being p-groups.

Perhaps the statement about pq can be salvaged: a group of order pq is a minimal non-primary-group, as in every subgroup is a r-group for some prime r. Every minimal-non-primary-group is solvable (by Redei's classification of minimal non-nilpotent or Burnside's pq theorem). It seems plausible that every minimal non-primary-group is of order p^n q with either n=1 or the Sylow p-subgroup elementary abelian and normal with n the order of p mod q. Minimal-non-P groups for various properties P have been studied, so it probably would be easy to find a reference for whatever is true. Even without giving the classification, one could give examples of groups (if n is the order of p mod q, then GF(p^n) contains a primitive q'th root of unity, and its action on GF(p^n) produces a semidirect product all of whose subgroups are p-groups or q-groups).

At any rate, I don't see a simple, clean way to copyedit the new contribution, and the article itself is pretty stubby so it is hard to bring it into line with some overarching theme. JackSchmidt (talk) 15:33, 11 December 2007 (UTC)[reply]

A group

The article A group is already in the more-than-five-days backlog of Wikipedia:Requested moves. Michael Slone (talk) 01:58, 14 December 2007 (UTC)[reply]

Divergent infinite products

I'm glad you noticed that I did not claim the infinite product was convergent, but only that the partial products converged to some limit. I just wanted to correct the claim that convergence to a non-zero value was required for definedness. After all, otherwise we would need to exclude the zeros from the domain of the identities given by the Weierstrass factorization theorem and the well-known product representations of functions, such as for the sine function! --Lambiam 20:10, 14 December 2007 (UTC)[reply]

Long Lake (Alpena, Michigan)

Jack: Thanks for the prompt assistance. 7&6=thirteen (talk) 00:23, 16 December 2007 (UTC)Stan[reply]

Grayling, Michigan

Jack:

I'm having problems with one of the road signs in this article. Could you help? Thanks. 7&6=thirteen (talk) 21:38, 17 December 2007 (UTC)Stan[reply]

Thanks for straightening that out. I was encountering some items where I didn't have a template to work from. Now I do. —Preceding unsigned comment added by 7&6=thirteen (talk • contribs) 22:03, 17 December 2007 (UTC)[reply]

No problem. I think I fixed the image link, and two text page links. I also learned I have been mistaken for over 30 years about the name of central in Dallas, TX, I75 versus US75. JackSchmidt (talk) 22:06, 17 December 2007 (UTC)[reply]

Talk:Unitary group#Other fields

Hi Jack,

Sorry for taking so long getting back (been busy); I've incorporated some of your and Rob's discussion into Unitary group#Generalizations. I've largely followed your outline, first mentioning indefinite forms, then the case of finite fields, then some very sketchy comments on over other algebras and algebraic groups (these latter could be fleshed out; I've little expertise on this). Hope this helps, and thanks for drawing me in!

Nbarth (talk) 00:15, 19 December 2007 (UTC)[reply]

Hi Jack,

I've made a further change to Unitary group#Generalizations, mentioning the connection to groups of Lie type and the interpretation of the unitary group as a Steinberg group (Lie theory).

Nbarth (email) (talk) 02:07, 2 January 2008 (UTC)[reply]

Re: triality: I did mention it in passing in the lede to Unitary group#Generalizations, though it could warrant a subsection. I've also connected ³D₄ to and from triality.

Also, thanks for the fix to Steinberg group (K-theory); I'd been puzzling through the texvc myself.

Nbarth (email) (talk) 05:40, 2 January 2008 (UTC)[reply]

Brauer–Nesbitt theorem BALETED

Yabbut, see the next diff, in which the non-redirecty article is added. Michael Slone (talk) 01:01, 19 December 2007 (UTC)[reply]

Re:Mathbot

I replied on my talk page. Oleg Alexandrov (talk) 04:28, 19 December 2007 (UTC)[reply]

Gaylord, Michigan

Having a problem putting in the picture for I-75 BL 7&6=thirteen (talk) 14:38, 20 December 2007 (UTC)Stan[reply]

Thanks. 7&6=thirteen (talk) 19:49, 20 December 2007 (UTC)Stan[reply]

AWB

The instructions say you won't be notified if you're approved to use AutoWikiBrowser, but I'm notifying you anyway. :) You're good to go now. --Elkman ^(Elkspeak) 17:39, 20 December 2007 (UTC)[reply]

Otsego County, Michigan

Having a problem putting in the picture for I-75 BL. Same as in Gaylord, Michigan. 7&6=thirteen (talk) 18:12, 20 December 2007 (UTC)Stan[reply]

I didn't find a good link for all the business loops grouped together, and the business loops are given very little attention on Interstate 75 in Michigan (but enough I could use as a model). There is a longer list article List of Business Routes of the Interstate Highway System#Interstate 75 that was also useful. I ended up just linking the image and the local article, Interstate 75 Business (Gaylord, Michigan). Feel free to change the link labels, or add back in an I-75 link. JackSchmidt (talk) 19:28, 20 December 2007 (UTC)[reply]

Thanks. Good job! I had all this nice new text and the images, and it was like having a scratch on the hood of my new Cadillac. 7&6=thirteen (talk) 19:36, 20 December 2007 (UTC)Stan[reply]

Cyclically reduced word

Do you think that there needs to be an article just to define the term cyclically reduced word? Michael Slone (talk) 23:10, 20 December 2007 (UTC)[reply]

It should establish notability and provide context. It is a notable concept, but probably should be included in a larger article, and this one made a redirect. If a relation is not cyclically reduced, then it can be shortened. In almost all cases this is a wise idea. In Todd-Coxeter enumeration one often includes all cyclic permutations of a word for improved performance.

There are too many dictionary definitions, and not enough articles. I could turn this one into a decent article, so I don't think it deserves deletion, but it'd be much easier to turn it into a good section of a broader article. JackSchmidt (talk) 23:25, 20 December 2007 (UTC)[reply]

Pewabic Pottery

Hi Jack. I would like to say: good work! —Preceding unsigned comment added by 86.151.154.226 (talk) 00:20, 21 December 2007 (UTC)[reply]

Jack:

No need to genuflect or explain. I'm not wedded to one set of doing this, and certainly not to one word. I appreciate your willingness to get involved. Carptrash had worked on this article for quite some time. I've only been playing with it a little over a month, although there are like 250 edits -- I've put in a lot of sources, and I'm not doing a big text thing somewhere else, but have been commuting from sources back to the article, etc., as it was layered together. You know the process, I'm sure.

I think that the main problem (as I elucidated on the discussion page) is that it is actually pretty short. In that context, we come to the conclusions, and lay them out without a lot of elucidation.

Moreover, we've just left out a LOT of the story, including the cast of characters, the course of action, the drama of the development of the art. I think that if we could marshall more facts, then the conclusions become more obvious, and we can let the readers do that on their own. However, it will require more text. I vote for that, FWIW. But doing that takes some research and some time. We actually have enough raw materials to do that -- I pulled together some really good articles that are cited in our text -- but it would require a lot of reading, a lot of thought, and then some slicing, dicing, cutting and pasting.

Ultimately, I think that this project is important. Among other things, I already had a disagreement with TeapotGeorge in Pottery, who is of the opinion that Pewabic Pottery has no importance to him, was never a big producer (he doesn't seem to recognize that quality can be just as important as quantity, maybe more so) and that he thinks they are (my words) a second-rate shop that is only marginally important and of interest to the "American Arts and Crafts" movement. He won't even recognize that it was an International movement. I don't know that we'll ever convince him (although he seems to be running Pottery like a tsar (even though he's from Columbia), but I think that the work that was (and is being) done at Pewabic has real artistic significance, and deserves to be highlighted and recognized.

I am glad that the article is getting some attention and needed help.

Best regards. 7&6=thirteen (talk) 02:50, 21 December 2007 (UTC)Stan[reply]

Additive number theory

To answer your question, compare the article Additive number theory, which is "too short to provide more than rudimentary information about [its] subject" (hence a stub), with the decidedly non-stubby article Number theory. Michael Slone (talk) 06:04, 21 December 2007 (UTC)[reply]

I just meant why was there only a stub article on additive number theory, one of the main branches of number theory, one of the main branches of mathematics. It'd be like having an encyclopedia where the graph theory article was still a stub. JackSchmidt (talk)

Ah, misunderstood. Speaking of graphs, have you ever seen "crystal" used in the sense of quiver? Some of the old revisions of Crystal (mathematics) seem to use it this way. (I can't tell what the new revisions say.) Michael Slone (talk) 06:14, 21 December 2007 (UTC)[reply]

I read some online notes once about crystals. It was a representation theory of supergroups thing, I think. Probably the source of the name is from Lusztig, from amazon:

...the theory of "crystal bases" or "canonical bases" developed independently by M. Kashiwara and G. Lusztig provides a powerful combinatorial and geometric tool to study the representations of quantum groups.

JackSchmidt (talk) 06:19, 21 December 2007 (UTC)[reply]

No luck finding the notes. It definitely used quivers to describe some ideas vaguely related to modular representation theory of finite simple groups. It was screwed (q-ewed?) up though and turned out completely useless. They were course notes though and easy to read as such things go. JackSchmidt (talk) 06:33, 21 December 2007 (UTC)[reply]

Northern Michigan

Jack: I extensively rewrote this article. However, the "See also" list is extremely long, and would be better if it were in two columns or more. I don't know how. Can you help?

Thanks 7&6=thirteen (talk) 14:43, 21 December 2007 (UTC)Stan[reply]

I made two edits so you could see which you prefer. The first one only works in FireFox (and lots of other less common browsers, but not IE7). It is very easy to tweak, though. You just add <div style="-moz-column-count:4; column-count:4;"> to the top and </div> to the bottom, where you can replace "4" (both times) with how many columns you want. 4 happened to be even, but 3 might be a better choice.

The other way, the way that works in IE7 and FireFox (and lots but not all less common browsers) is the {{multicol}} template. The only problem with it is you have to explicitly say where the columns break, so adding new items to the list is a pain in the butt. I suspect you rarely have to add cities, so I hope it was the right choice. It was the one I left on the article. Adjusting column lengths and widths just means moving, adding, or deleting those {{multicol-break}} tags.

Help:List has some information on doing this stuff, but leaves out the first solution. Math tends to use the first solution with the <div>, but probably because there are some commonly used math websites that worked better on FireFox for many years, and so everyone has FireFox. JackSchmidt (talk) 17:34, 21 December 2007 (UTC)[reply]

You've helped me again. And given me some valuable instructions, too. There is so much to learn.

It looks much better, I think. 7&6=thirteen (talk) 17:58, 21 December 2007 (UTC)Stan[reply]

Commercial link in Gaylord, Michigan

I note that 24.180.198.126 continues repeatedly to put in a link to a website that starts with the heading

"REAL ESTATE IS MY BUSINESS"

There is nothing else on that website. It is a commercial link, pure and simple. It does NOT belong in Wikipedia.

Moreover, he/she removes links to the Chamber of Commerce, which actually has lots of community information links, and to the airport. This is very strange.

Apparently this contributory wants to be the ONLY 'official Wikipedia endorsed real estate agent in Gaylord, Michigan.

It is wrong.

I don't know what to do, but would be happy to entertain any suggestions or assistance.

7&6=thirteen (talk) 18:42, 25 December 2007 (UTC)Stan[reply]

Alternation group

Thanks for correcting me. If I understand well the A₃ is isomorphic with the cyclic group of order 3. Is this correct? If so, I would like to put a note about this fact on the article. Cyb3r (talk) 13:12, 31 December 2007 (UTC)[reply]

No problem. As I mentioned in the edit summary, it appears to be a common confusion. Textbooks often explicitly prove A4 is not simple, and A5 is simple, and then An is simple for n≥5, but leave out A1, A2, A3 as being a little too small to say anything about. One might check out list of small groups to see how there are not very many small groups, so they can usually be checked by hand easily enough.

The alternating group A₃ is a group of order 3. Every group of order 3 is a cyclic group of order 3. Every cyclic group of order 3 is isomorphic to A₃. Sometimes one studies a particular cyclic group of order 3, the set {0,1,2} under addition modulo 3, also known as Z/3Z. This group is also isomorphic to A₃. JackSchmidt (talk) 18:17, 31 December 2007 (UTC)[reply]

Whitehead lemma steinberg etc.

Hi Jack,

I'm no expert on the group theory of the general linear group; I just stumbled into these from the unitary group → Steinberg groups, and suddenly found myself in (familiar!) algebraic K-theory land.

So I don't know what the derived subgroup of GL(n,A) looks like, for instance, or whether the group GL(A) determines the ring (all that comes to mind is that $Z(GL_{n}(A))=A^{*}$ , which isn't enough but isn't bad). I'm not in a university any longer, so I'm not in touch with people who'd know (and I'm not really sure whom to ask even then).

As you've presumably noticed, I've added some discussion of the connection between the Steinberg group and the special linear group. Abstractly the answer is given by this short exact sequence relating St and GL via K₁ and K₂; hopefully this answers your questions. (I don't know a presentation of SL in terms of transvections generally though.)

I do come from a geometric topology background, and before that algebraic geometry, and have a long-standing affair with algebraic topology, so I'm largely one of THEM, but I do like to understand what's going on (rather than quoting results), and I'm in no hurry, so I've the time to luxuriate (read: actually understand results).

I've added "stable" in Steinberg group where relevant; trust it looks saner now.

Isn't $\operatorname {E} _{n}(A)$ always perfect (for $n\geq 3$ ), due to the 2nd Steinberg relation (this struck you as insane)?

(I have no intuition for why kernel St → GL should be the center, but then, I have no intuition for St period: it's purely formal to me.)

Nbarth (email) (talk) 01:38, 6 January 2008 (UTC)[reply]

Thanks for the articles. They've been very helpful. No worries about the random questions.

The GL(A) determining A was phrased in the form "K_n(A) is an invariant of GL(A) for positive n, how about for n=0", so it might still be interesting to you (though I still have no idea where to look). For non-commutative A, M_n(A) does not determine A, but does up to Morita equivalence. For commutative rings, Morita equivalence is just ring isomorphism, so I think it should be fine to add "A commutative" or "up to Morita equivalence". However, Units(R) does not determine R, and so GL_n(R) = GL_n(S) does not even imply that M_n(R)=M_n(S). I could not find any examples where this held for all n, so I am not sure if "stable" magically changes things here like it usually does.

"Isn't

\operatorname {E} _{n}(A)

always perfect (for

n\geq 3

), due to the 2nd Steinberg relation (this struck you as insane)?"

Perfection seems fine, just the idea that the kernel is the center. The "second center" of a perfect group is equal to the center. Saying that the kernel of St(A) to E(A) is the center of St(A) should imply that the center of E(A) is trivial, which, as far as I know, it is not. If St/Z(St) = E and St is perfect, then Z(E)=Z^2(St)/Z(St)=Z(St)/Z(St)=1. This doesn't seem kosher for A=Z/5Z for instance, where I think E_n=SL_n is perfect, and often has nontrivial center.

Probably my mistake is in thinking Z(E_oo) is nontrivial, since Z(E_n(A)) is repeatedly (in n) trivial. JackSchmidt (talk) 23:39, 6 January 2008 (UTC)[reply]

Second center / Higher centers

Hi Jack, I hadn't heard about the second center, but I figured I understood what you meant, so I wrote it up at Center (group theory)#Higher centers, together with notes about centerless groups and perfect groups. I don't know any references or uses or further information about what I've termed higher centers (is that the correct term?).

Oh. I guess you noticed.

Nbarth (email) (talk) 01:23, 9 January 2008 (UTC)[reply]

"Higher center" is definitely common in spoken math. I think symbols Z^i or "upper central series" may be more common written, but I don't have sources nearby to check. The section looks nice. I linked to upper central series, but that is just a redirect. I can't decide if every little group theory definition needs its own article. I think User:Zundark is improving the Derived length / Derived series situation for soluble groups, so I've avoided thinking about what to do for central series of nilpotent groups. I like your merge of Gruen's lemma into perfect group. I'm pretty sure he has some more important lemmas that could be added eventually, but until then, that article was going to stay pretty stubby. JackSchmidt (talk) 01:29, 9 January 2008 (UTC)[reply]

Glad you liked them!

Agreed re: how finely to divide articles—I was thinking this for lower central series/upper central series, which clearly have interest beyond defining nilpotent group! I mentioned the analogy in Lie algebras, and I've actually used this: the lower central series of a Lie algebra is crucial in understanding higher linking, meaning phenomena like Borromean rings! (Borromean rings have an algebro-topological invariant in the second LCS quotient; it can be expressed as a Massey product.)

Nbarth (email) (talk) 01:39, 9 January 2008 (UTC)[reply]

If lower central series gets its own article, then one can talk a bit about how the intersection of the lower central series of a free group is trivial and that the quotients are free abelian with an explicit basis given by Hall's basic commutators, a result carrying over directly to Lie algebras. Lie algebras are used fairly often for p-groups, since for nilpotent groups of nilpotency class c which are p-torsion-free for p <= c, the Baker-Campbell-Hausdorff formula lets one define the Lazard correspondence between these "Lie" groups and their corresponding Lie rings. The Lie ring for a p-group has underlying abelian group the direct sum of the lower central factors, and the Lie bracket is just the Lie bracket moving down the expected number of terms. The way lower central series is stuck inside nilpotent group makes things like this a bit too long of an aside. JackSchmidt (talk) 01:52, 9 January 2008 (UTC)[reply]

I didn't quite follow that last bit...perhaps now there's a suitable page to elaborate?

Nbarth (email) (talk) 00:17, 16 January 2008 (UTC)[reply]

Glad you enjoyed the recent hypercenter/higher centers edits; I just edited higher centers and discussed transfinite recursion, and relegated "you need to use transfinites for the hypercenter!" to a footnote.

Nbarth (email) (talk) 00:21, 16 January 2008 (UTC)[reply]

There is definitely room now. The free group part has been added. The "use of Lazard correspondence in p-groups" has not, but if you are impatient, Khukhro's book on automorphisms of p-groups is a good reference for this. You might like Segal's book on polycyclic groups where the similar idea is called Malcev correspondence and used in characteristic 0. I think in Lie groups it is just called "log" and "exp" or something. JackSchmidt (talk) 12:58, 16 January 2008 (UTC)[reply]

UCS : Hypercenter :: LCS : ??

So the limit of the UCS is the hypercenter, and the limit of the derived series is the perfect core, which suggests the question: what's the limit of the LCS?

Nbarth (email) (talk) 00:18, 16 January 2008 (UTC)[reply]

For a finite group the limit of the LCS is the nilpotent residual, the second largest term in the lower Fitting series. The book of Doerk and Hawkes should have information in the finite case. I don't know if it has a name when the length is more than ω, but one might call γ_ω(G) the residually nilpotent residual, or just the nilpotent residual for short. It is the smallest subgroup with residually nilpotent quotient. JackSchmidt (talk) 12:52, 16 January 2008 (UTC)[reply]

Schur–Weyl, Schur's lemma

Hello, thanks for your note and for your consideration. Feel free to amend Schur–Weyl duality with more examples, especially if you can make them clear. The case k = 3 has an additional difficulty in that one has to consider separately n = 2 (since the expansion is cut off at min(k,n) rows of D). If I have time, I'd add n=2 and all k, just to demonstrate what happens in the "opposite" case k ≥ n when eventually all polynomial representations of GL_n appear, but only a subset of representations of S_k.

Thank you also for correcting my carelessness at Schur's lemma. However, I don't quite like you revision for the following reason: it starts with unnecessary generality (indecomposable modules), whereas we should explain the plain Schur's lemma for finite-dimensional representations of groups first, and only then talk about various extensions. I am a bit surprised actually at the order of you presentation, in light of the very sensible comments you've made about being accessible to non-specialists. Best, Arcfrk (talk) 23:32, 20 January 2008 (UTC)[reply]

@@ Line 281: / Line 281: @@
 :For a finite group the limit of the LCS is the '''nilpotent residual''', the second largest term in the lower [[Fitting length|Fitting series]]. The book of Doerk and Hawkes should have information in the finite case.  I don't know if it has a name when the length is more than ''ω'', but one might call ''γ''<sub>''ω''</sub>(''G'') the residually nilpotent residual, or just the nilpotent residual for short.  It is the smallest subgroup with residually nilpotent quotient. [[User:JackSchmidt|JackSchmidt]] ([[User talk:JackSchmidt#top|talk]]) 12:52, 16 January 2008 (UTC)
+== Schur–Weyl, Schur's lemma ==
+Hello, thanks for your note and for your consideration. Feel free to amend Schur–Weyl duality with more examples, especially if you can make them clear. The case ''k'' = 3 has an additional difficulty in that one has to consider separately ''n'' = 2 (since the expansion is cut off at min(''k'',''n'') rows of ''D''). If I have time, I'd add ''n''=2 and all ''k'', just to demonstrate what happens in the "opposite" case ''k'' &ge; ''n'' when eventually all polynomial representations of ''GL''<sub>''n''</sub> appear, but only a subset of representations of ''S''<sub>''k''</sub>.
+Thank you also for correcting my carelessness at Schur's lemma. However, I don't quite like you revision for the following reason: it starts with unnecessary generality (indecomposable modules), whereas we should explain the plain Schur's lemma for finite-dimensional representations of groups first, and only then talk about various extensions. I am a bit surprised actually at the order of you presentation, in light of the very sensible comments you've made about being accessible to non-specialists. Best, [[User:Arcfrk|Arcfrk]] ([[User talk:Arcfrk|talk]]) 23:32, 20 January 2008 (UTC)