From: baez@math.galaxy.edu (John Baez) Subject: Re: The Monster and the Leech Date: 3 Oct 2000 21:53:50 GMT Newsgroups: sci.physics.research Summary: [missing] In article <20000930000618.18144.00000214@ng-bd1.news.gateway.net>, Jim Heckman wrote: >John Baez wrote: >>I don't understand all this stuff (what's a Schur multiplier?) >Sort of the inverse of the outer automorphism group, kind of. Every >finite perfect group (i.e., those equal to their own commutator >subgroup, which of course includes all non-abelian simple groups) >has a unique perfect "maximal central extension" = "univeral >covering group", whose center is an abelian group (duh!) -- the >Schur multiplier -- whose structure says something interesting about >the structure of the original perfect group. Hmm. Jim Dolan and I were studying group cohomology last spring and we ran into this type of idea without knowing this "Schur multiplier" terminology. We got a bit confused for a while and then I think we got straightened out, as follows: We were actually thinking about extensions rather than central extensions. Extensions of a group G by an abelian group A are classified by the 2nd cohomology group H^2(G,A). So we thought that maybe the 2nd homology group H_2(G,Z) would serve as the "universal extension of G". What we meant by this was that an extension of G by A was the same as a homomorphism from H_2(G,Z) to A. Or, more precisely, that there's an isomorphism: H^2(G,A) = Hom(H_2(G,Z),A) But this was wrong, in general. The universal coefficient theorem says there's an extra term: H^2(G,A) = Hom(H_2(G,Z),A) + Ext(H_1(G,Z),A) I forget if we ever came up with a snappy description of the universal property of H_2(G,Z) - i.e., a snappy description of what a homomorphism from H_2(G,Z) to A amounts to. But now, reading Rotman's "Homological Algebra" (which I carefully avoided while studying this stuff with Jim, since we wanted to reinvent everything ourselves), I see that he calls H_2(G,Z) the "Schur multiplier" of G, and states that when G is perfect, this is the universal *central* extension of G. And for more general G, H_2(G,Z) has something to do with something called "stem covers", whatever those are. I'm getting a little confused, as I often do, about the role of extensions vs. central extensions in the grand scheme of homological algebra. Having finally convinced myself that H^2(G,A) classifies *extensions* of G by A, I find it terrifying that H_2(G,Z) is sometimes the universal *central extension* of G. My last shred of sanity relies on the fact that this only happens when G is perfect, and that in general, H_2(G,Z) has some other meaning. ============================================================================== From: mckay@cs.concordia.ca (MCKAY john) Subject: Re: The Monster and the Leech Date: 4 Oct 2000 00:09:11 GMT Newsgroups: sci.physics.research In article <8rdkhe$ag1$1@Urvile.MSUS.EDU>, John Baez wrote: >In article <20000930000618.18144.00000214@ng-bd1.news.gateway.net>, >Jim Heckman wrote: >>John Baez wrote: >>>I don't understand all this stuff (what's a Schur multiplier?) >>Sort of the inverse of the outer automorphism group, kind of. Every >>finite perfect group (i.e., those equal to their own commutator >>subgroup, which of course includes all non-abelian simple groups) >>has a unique perfect "maximal central extension" = "univeral >>covering group", whose center is an abelian group (duh!) -- the >>Schur multiplier -- whose structure says something interesting about >>the structure of the original perfect group. >Hmm. Jim Dolan and I were studying group cohomology last spring and >we ran into this type of idea without knowing this "Schur multiplier" >terminology. We got a bit confused for a while and then I think we >got straightened out, as follows: [BIG SNIP] The Schur multiplier is the second cohomology group, H^2(G,C^*) with trivial action on C^*. It is the group of "factor sets". If G is perfect then it is unique. See Curtis & Reiner (old book) on Representation Theory. An old observation of mine ( Santa Cruz Conference Proceedings on Finite Groups, AMS SYmp. Pure Math, vol 37, (1980), page 183) is that the sporadic simple groups, M, Baby, F24 have Schur multipliers = fundamental group (connection number for physicists) of E8,7,6 respectively, namely C1,C2,C3. I believe this to be a deep observation (it remaining unexplained for the past 20 years!). I expect del Pezzo surfaces to be involved. Borcherds' plenary address to ICM (1998) Berlin address mentions this. John McKay -- But leave the wise to wrangle, and with me the quarrel of the universe let be; and, in some corner of the hubbub couched, make game of that which makes as much of thee.