User talk:JackSchmidt/Archives/2010/01

Combinatorial description Coxeter complex
Hello,

I assume you are also the Jack Schmidt from ask an algebraist? You have answered my questions several time there (I use another name there), and I thought that it would be polite if I return the favour by answering (partially) a question I have asked before.

The question was: how to understand the Coxeter complex of a Coxeter group? So for each maximal parabolic subgroup $$H$$, there is a certain type of objects (explicitly: its cosets), and two objects are incident if the corresponding cosets $$g_1 H_1$$ and $$g_2 H_2$$ are not disjoint.

This should be the answer: make a system of simple roots. Choose any node of the diagram, and take a vector in the 1-dimensional orthogonal complement of all other simple roots. Its orbit of the reflection group corresponds with the objects of that type.

For instance, for F4, one can take as simple roots: $$e_1-e_2, e_2-e_3, e_3$$, $$-1/2 (e_1+e_2+e_3+e_4)$$ where $$e_i$$ denotes a four-dimensional orthonormal basis.

For the leftmost node, we obtain this vector in the orthogonal complement :$$e_1-e_4$$. Its orbit consists of all 24 vectors $$\pm e_i\pm e_j$$.

For the other end node, we obtain this vector in the orthogonal complement: $$e_4$$. Its orbit consists of all 8 vectors $$\pm e_i$$ and all 16 vectors $$(\pm e_1\pm e_2\pm e_3\pm e_4)/2$$.

Unfortunately, I haven't quite figured out how to work out incidence, but it is probably seen in terms of the inner product. If you are interested, I will let you know as well when I find out.

Kind regards, evilbu. —Preceding undated comment added 2009-11-26T08:01:04Z.


 * Thanks. Certainly, I am interested (and I suspect you are right about some simple condition on the inner product).  I've been fascinated by coset geometries since reading Aschbacher's Finite Group Thoery textbook.  I am currently studying the geometry of the classical groups, since I think coset geometries are motivated by these classical examples.  I'm reading Barwick and Ebert's "Unitals in Projective Planes"  right now, and it has gone the smoothest so far of all the finite geometry books I've read.  The only example I have from multiple sources is that GL on a Borel subgroup is the classical projective geometry.  However, this example has never been very convincing for me.  I think these "ovals" and "quadrics" and things explain lots of the low dimensional Chevalley groups, which hopefully should be enough variety.  Actually, I also found polydrons really helpful for 3-dimensional real geometry, which might be more relevant for the Coxeter complexes. JackSchmidt (talk) 17:22, 26 November 2009 (UTC)


 * Hello, I thought it might be helpful to inform you of some other things. First of all, I figured out how the inner products work.  You don't just have to calculate the orthogonal complement, you have to calculate the fundamental weight: the unique vector orthogonal to all but one, and with inner product one with the remaining one.  Having calculated those fundamental weights actually gives you what is known as a fundamental chamber: in the case of F4, a set of four objects, one of each type, and all incident with each other.


 * What a coincidence that you are reading that book by Barwick and Ebert! I have also read some parts of it, and I agree that it is very clear, not only on the unitals but on geometry in general.


 * My background is actually the other way around: I have been doing (finite incidence) geometry for at least four years now, but I am very slowly also becoming familiar with group-theoretic concepts. Concerning your comments on the projective geometry, I'm not sure if I understand it completely, so forgive if I'm saying something you know very well.. but it is in fact not hard to create geometries related to other finite coxeter groups. You already mentioned the projective geometries, related to A_n.  There also the polar spaces, related to the Coxeter group B_n.  They are all obtained by taking totally isotropic subspaces in a vector space with respect to a quadratic or bilinear form.  A fundamental difference with the projective geometries is that there in fact several constructions for one field. For instance for B_3, you can take the 1-dimensional, 2-dimensional and 3-dimensional subspaces on the quadric $$x_0^2+x_1 x_2+x_3 x_4

+x_5 x_6=0$$, but you can also take the 1-dimensional, 2-dimensional and 3-dimensional subspaces on the quadric$$ x_0^2+x_1 x_2+x_3 x_4+F(x_5,x_6)=0$$, where F is some irreducible homogeneous polynomial of degree two. In the first case, there are $$(q^6-1)/(q-1), (q^6-1)(q^2+1)/(q-1)$$ and $$(q+1)(q^2+1)(q^3+1)$$ subspaces of dimension 1,2,3 (respectively), and in the case the numbers are $$(q^4+1)(q^2+q+1), (q^4+1)(q^2+q+1)(q^3+1)$$ and ($$q^2+1)(q^3+1)(q^4+1)$$. Note however that in both cases, if you plug in q=1, you get the numbers for a octahedron: 6 points, 12 lines and 8 faces. I often hear people referring to projective and polar spaces as "the classical ones", because they can be explained in "human language" to students. The enormous size is also a problem with those other, more abstract geometries. For instance: for F4, you get a geometry with $$(q^4+1)(q^{12-1})/(q-1)$$ points (which becomes just 24 if q=1).

Kind regards, Evilbu (talk) 21:55, 30 December 2009 (UTC)


 * I am finding Stephen D. Smith's book on Subgroup Complexes just about perfect. Instead of parabolic subgroups, you look at unipotent subgroups, and yet it gives the same space up to homotopy.  Then you can look at arbitrary finite groups by looking at the p-subgroups, which is quite convenient as I've already been studying those under the name of (Puig's) Frobenius categories.  Benson and Smith also have a fairly reasonable book, "Classifying Spaces of Sporadic Groups" or so that discuss these various geometries. JackSchmidt (talk) 01:49, 23 January 2010 (UTC)

Reference: Normal subgroups of prime index?
A delightful fact I observed in grad school is that the normal subgroups of prime index in a group are a projective space, rather naturally. I’ve written this up at Index of a subgroup: Normal subgroups of prime index (this edit), and while it’s a pretty simple remark, I don’t have a reference to hand (elementary group theory books tend to steer clear of geometry, other than representation theory, and I don’t know the non-elementary group theory books) – would you know one? Thanks!
 * —Nils von Barth (nbarth) (talk) 07:56, 27 January 2010 (UTC)


 * The short answer is no, I don't know of a reference that talks about things this way. The long answer is that your observation factors into two famous and citable observations, but I don't think too many people would want to "multiply" them back together.
 * (1) The normal subgroups of G of index p intersect in a subgroup called Ep(G). G/E^p(G) is an elementary abelian p-group (and the largest elementary abelian p-group onto which G surjects). E^p(G) has many incredibly interesting properties, for instance G/E^p(G) ≅ P/(P∩(P^p[G,G])) via the transfer.  E^p(G) has important subgroups (in terms of G): A^p(G) is the intersection of the p-power index subgroups of G containing [G,G], and O^p(G) is the intersection of the p-power index normal subgroups of G.  G/A^p(G) ≅ P/(P∩[G,G]) and P∩[G,G] is called a focal subgroup of G and is determined from the local structure of G, P∩[G,G] = < [x,g] : x in Q ≤ P, and g in N_G(Q) >.  G/O^p(G) ≅ P/(P∩γ∞(G)) and P∩γ∞(G) is a hyperfocal subgroup of G (where γ∞(G) is the nilpotent residual of G, the intersection of the lower central series of G).  The hyperfocal subgroup is determined by local information of G, P∩γ∞(G) = < [x,g] : x in Q ≤ P, and g in O^p(N_G(Q)) >.  Focal, focal, blah, blah, tons of information on it, and its relation to other pieces of the structure of G.  I put some of this in focal subgroup theorem.
 * (2) Elementary abelian groups are affine geometries. Their maximal subspaces form the points of the associated (dual) projective geometry.  This is very standard "geometry via linear algebra".
 * I personally would never combine these two observations because of the critical problem that G does not act on the geometries G/A^p(G) or P*(G/A^p(G)), where P*(V) is the projective geometry on the vector space V defined by taking n-1 dimensional subspaces as points and n-k dimensional subspaces as the k-1 dimensional facets. Now G might have lots of elementary abelian sections, maybe even some normal elementary abelian sections, and the action of G on those sections may not be trivial, and so might be interesting.  However, G acts trivially on the vector space G/A^p(G) and all of its associated geometries.  Even looking at "partial actions" on P/(P∩[G,G]), the point is that the focal subgroup is defined precisely so that G is forced to act trivially on any facet that it doesn't shatter (by sending it outside of P). JackSchmidt (talk) 20:55, 27 January 2010 (UTC)


 * Thanks for the thoughtful and detailed response – that’s a much deeper structure than my naive observation, and good point about “no action”. I’ll work to incorporate and digest those at the relevant pages so people get a better perspective. (To start, I’ve made upper p-series and lower p-series and linked them so they are more discoverable)
 * —Nils von Barth (nbarth) (talk) 02:03, 28 January 2010 (UTC)


 * Thanks Jack – I’ve read up some and had a go at making Focal subgroup theorem a little easier-to-scan, and making Index of a subgroup actually point people in the right direction for group theory, though still noting the cute fact that you can’t have exactly two subgroups of index 2. (Also, Category:p-groups is now much better populated.)
 * (Hope I haven’t introduced any errors in the process!)
 * As you indicate, for the study of group structure you really want to look deeper at the p-residual subgroup (and fusion, focal, etc.), and that this geometry, while cute, is really just scratching the surface and isn’t saying much about the group.
 * —Nils von Barth (nbarth) (talk) 08:39, 28 January 2010 (UTC)


 * Thanks. I think your organization of those subgroups is much better for the encyclopedia, though I'm glad my wandering style led someone somewhere nice.  I noticed R.A. Wilson's textbook is now published:
 * There is also a DOI per chapter if you wanted to cite them that way. JackSchmidt (talk) 15:17, 28 January 2010 (UTC)
 * Thanks (and good on Rob)! (I’ve updated the refs.)
 * —Nils von Barth (nbarth) (talk) 06:17, 29 January 2010 (UTC)
 * —Nils von Barth (nbarth) (talk) 06:17, 29 January 2010 (UTC)