User talk:JackSchmidt/Archives/2008/01

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)

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)

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)

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)

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)

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)

"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)

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)

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)

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)

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)

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)

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)

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)

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)

Teresa Bright

(replied to my post)

Jack, can you take a look at the content again and let me know which additional changes you would like to see? Dondt1 30 January 2008

Sure. I think it may still be too close to the source, but it has been a week and all the other articles nominated at that time have been deleted, so I'll take down the banner for now. Probably your best bet is to find a second source. I myself tried to reword the article, but honestly most of it is straightforward facts. If one had a second source it would be easy to be more creative in presentation. JackSchmidt (talk) 00:00, 31 January 2008 (UTC)