April 19, 2002

This Week's Finds in Mathematical Physics (Week 180)

John Baez

First, a news flash: they may have found a quark star... or two!

In case you're wondering, a "quark star" is a hypothetical entity smaller and denser than a neutron star, where the pressure is so high that the neutrons get crushed into a mess of quarks. Nobody really knows if this is possible without the darn thing collapsing all the way into a black hole. However, if it happened, a bunch of the down quarks in the neutron would turn into strange quarks, which are somewhat more massive, but energetically favored nonetheless in situations like this where the Pauli exclusion principle reigns supreme. Folks refer to this phenomenon as "strangeness enhancement". It sounds like some sort of surgical operation undergone by Michael Jackson, doesn't it? But anyway, for this reason, quark stars are also known as "strange stars".

Back in "week117" I described evidence for strangeness enhancement in quark-gluon plasma experiments at Brookhaven and elsewhere, but it would be really cool to see it in nature. People have been looking for quark stars for some time, with no success, but NASA has just announced the discovery of two entities that might be quark stars:

1) Cosmic X-rays reveal evidence for new form of matter, http://www1.msfc.nasa.gov/NEWSROOM/news/releases/2002/02-082.html

The Chandra X-ray observatory (see "week143" for info on this marvelous satellite) has seen two stars, romantically entitled RXJ1856 and 3C58, that look sort of like neutron stars... but apparently too small or too cool to be neutron stars! There's always the possibility that something else is going on, but folks are thinking they look like strange stars. Stay tuned.

Okay, now for some math. First some news on topos theory, and then I'll return to the theme of "week178": Lie groups and geometry... leading up to a taste of twistors.

Peter Johnstone is a category theorist who can often be seen playing backgammon in the common room of the Department of Pure Mathematics and Mathematical Statistics at Cambridge University. He also selects the wines at St. Johns. But he must have been working dreadfully hard for the last decade or so, because he's produced a book of mammoth proportions:

2) Peter Johnstone, Sketches of an Elephant: a Topos Theory Compendium, Cambridge U. Press. Volume 1, comprising Part A: Toposes as Categories, and Part B: 2-categorical Aspects of Topos Theory, 720 pages, to appear in June 2002. Volume 2, comprising Part C: Toposes as Spaces, and Part D: Toposes as Theories, 880 pages, to appear in June 2002. Volume 3, comprising Part E: Homotopy and Cohomology, and Part F: Toposes as Mathematical Universes, in preparation.

I can't wait to dig into this. A topos is a kind of generalization of the universe of set theory that we all know and love, but topos theory is really a wonderful way to unify and generalize vast swathes of mathematics - you could say it's the way that logic and topology merge when you take category theory seriously. I've really just begun to get a glimmering of what it's all about, so I'm curious to see Johnstone's overall view of the subject.

If you're wondering what a topos actually is, and you're too impatient to wait for Johnstone's books to come out, I suggest that you start with my quick online summary:

3) John Baez, Topos theory in a nutshell, http://math.ucr.edu/home/baez/topos.html

and then try the books I recommended in "week68", along with this one:

4) Colin McLarty, Elementary Categories, Elementary Toposes, Oxford University Press, Oxford, 1992.

which I only learned about later, when McLarty sent me a copy. I wish I'd known about it much sooner: it's very nice! It starts with a great tour of category theory, and then it covers a lot of topos theory, ending with a bit on various special topics like the "effective topos", which is a kind of mathematical universe where only effectively describable things exist - roughly speaking.

Now, in "week178" I described some things James Dolan and I were learning about Lie groups and geometry. In the meantime we've learned so much that I sort of despair of conveying it all... beautiful, wonderful stuff! We're even beginning to understand the theory of "buildings", which I had long considered an impenetrable bastion of incomprehensibility.

But instead of rhapsodizing, let me dive in and explain as much as I can. Last time I explained that every complex simple Lie group G gives rise to a generalization of projective geometry. When we take G = SL(n,C) we get ordinary projective geometry, and I focussed on this case, but I described how things work in general. Today I want to dig a little deeper into the general theory then consider a bunch of examples, leading up to Penrose's theory of twistors.

First, remember how this game goes. Every complex simple Lie group G has a bunch of maximal solvable subgroups, all basically the same as each other - so people pick one and call it "the Borel subgroup", or B for short. When G = SL(n,C) we can take B to be the subgroup of upper triangular matrices. When doing geometry with some symmetry group G, any subgroup should be thought of as the group of transformations that preserves some sort of "figure" - some geometrical object. The importance of the Borel subgroup is that it preserves a "maximal flag". For G = SL(n,C) acting on complex projective space, this is just:

a point lying on a line lying on a plane lying on a 3-space lying on...

Having chosen a Borel subgroup B, there is a finite set of subgroups containing B and smaller than G - people call these "parabolic subgroups". These preserve all the various smaller kinds of flag, which in the SL(n,C) case are things like

a point lying on a plane lying on a 5-space

a line

The parabolics range in size from B to the "maximal parabolic subgroups". The bigger the subgroup, the less it preserves, so the maximal parabolics preserve the simplest flags, like "a point", "a line", "a plane", and so on. In this case the flag manifold G/P is usually called a "Grassmannian".

Now, the cool part is that you can read off the parabolic subgroups from the Dynkin diagram of a simple Lie group: they correspond to subsets of the dots! The maximal parabolics correspond to the dots themselves. For SL(n,C) it works like this... I'll illustrate with the case n = 4:

                           o------o------o
                        points  lines  planes       

                           x------o------x
                        points  lines  planes       

                               * * * *
                               0 * * *          
                               0 * * *
                               0 0 0 *

Even better, this game lets you get all the finite-dimensional irreducible representations of your complex simple group G. I'll say how it goes without explaining why it works. To get an irrep, just label each Dynkin diagram dot with a natural number! The subset of dots labelled by nonzero numbers determines a parabolic subgroup P. The numbers themselves pick out a complex line bundle over G/P. The group G acts on G/P, of course, and it also acts on this line bundle. Now, G/P is always a complex manifold since G and P are complex, so it makes sense to talk about holomorphic sections of this line bundle. The space of these forms a finite-dimensional irrep of G!

To really understand this deeply, you should learn a bit about geometric quantization. However, let's just assume it works and see what happens in some examples.

First consider G = SL(n,C). Here we've already seen that the maximal parabolics are the subgroups preserving various obvious figures in complex projective space:

                           o------o------o
                        points  lines  planes       

                           1------0------0

                           0------1------0

                           0------0------1

In general, the irreps we get by labelling one dot with a 1 and the rest with 0 are the most exciting: they're called the "fundamental" reps. In math jargon, they generate the representation ring of G. Even better, there's a simple recipe for taking a Dynkin diagram with dots labelled by numbers and finding the corresponding irrep inside a tensor product of symmetrized tensor powers of these fundamental reps, where the numbers labelling the dots tell you which powers to use. For SL(n,C) this is just the theory of Young diagrams, which I discussed in "week 157". So, we're just generalizing the heck out of that.

Even if you don't understand what I just said, you can rest assured knowing that we can completely master all the irreps of G once we figure out the fundamental ones. So, we'll focus on those.

We've more or less beat SL(n,C) to death, so let's see what happens with some other simple Lie groups... for example, the groups Spin(n,C). If you don't know these guys, first think about SO(n,C). This is the group of all linear transformations of Cⁿ preserving the symmetric bilinear form

x.y = x₁ y₁ + ... + x_n y_n

Unfortunately SO(n,C) is not simply connected, so not all reps of its Lie algebra give reps of the group. So, to get group representations from ways of labelling the Dynkin diagram by numbers, we need to work with its double cover, the "spin" group Spin(n,C).

You may be more familiar with the compact real forms of these groups. The compact real form of SO(n,C) is the good old rotation group in n dimensions, SO(n). The compact real form of Spin(n,C) is the double cover of SO(n), called Spin(n). The irreps of Spin(n,C) give unitary irreps of Spin(n), so you can think about them that way if you prefer.

The Dynkin diagram of Spin(n,C) looks really different depending on whether n is even or odd. It takes a while for the pattern to become clear - it's obscured by lots of delightful coincidences in low dimensions. I'll work through these low dimensions and then say the general pattern. If you're the sort who can't stand reading long lists of facts until you've seen the pattern they fit, jump ahead to where I talk about Spin(9,C) and Spin(10,C). I'm gonna climb my way up there slowly, taking my time to smell the flowers.

The Dynkin diagram of Spin(3,C) is just a single dot:

                      o

Spin(3,C) = SL(2,C).

The fundamental representation corresponding to the single dot in the Dynkin diagram is called the "spinor" representation of Spin(3,C): it's just the obvious rep of SL(2,C) on C². This fact is crucial for understanding spin-1/2 particles in 3d space.

The Dynkin diagram of Spin(4,C) is two dots, not connected by an edge:

                      o



  
                      o

Spin(4,C) = SL(2,C) x SL(2,C).

The Dynkin diagram of Spin(5,C) is two dots connected by a double edge:

                      o===>===o 

Spin(5,C) = Sp(4,C).

The fundamental rep corresponding to the left dot in the Dynkin diagram comes from the obvious rep of SO(5,C) on C⁵ - what physicists would call the "vector" rep. The fundamental rep corresponding to the right dot comes from the obvious rep of Sp(4,C) on C⁴ - it's called the "spinor" rep of Spin(5,C). This would be fundamental for studying spin-1/2 particles in 5-dimensional spacetime if anyone were interested... but not many people are.

The Dynkin diagram of Spin(6,C) has three dots:

                          o
                         /
                        /
                       /
                      o
                       \
                        \
                         \
                          o

Spin(6,C) = SL(4,C).

The fundamental rep corresponding to the left dot comes from the obvious rep of SO(6,C) on C⁶ - the "vector" rep again. The reps corresponding to the other dots are the left- and right-handed spinor reps of Spin(6,C), coming from the obvious rep of SL(4,C) on C⁴ and its dual. This is fundamental for understanding spin-1/2 particles in 6-dimensional space - for example, the 6 extra curled-up dimensions in string theory. And as we'll see, it's also basic to Penrose's theory of twistors!

At this point we're done with all the cute isomorphisms, so let us line them up and admire them before bidding them farewell:

Spin(3,C) = SL(2,C)
Spin(4,C) = SL(2,C) x SL(2,C)
Spin(5,C) = Sp(2,C)
Spin(6,C) = SL(4,C).

Spin(3) = SU(2)
Spin(4) = SU(2) x SU(2)
Spin(5) = Sp(2)
Spin(6) = SU(4).

The Dynkin diagram of Spin(7,C) has three dots:

                     o-------o===>===o 

The Dynkin diagram of Spin(8,C) has four dots:

                                  o
                                 /
                                /
                               /
                      o-------o
                               \
                                \
                                 \
                                  o

I've said a lot about triality in "week61", "week91" and elsewhere, but right now it's just a distraction - I'm trying to get you to see the pattern of Spin(n,C) Dynkin diagrams, and I'm hoping that by now it's apparent: an alternation between odd and even dimensions, and so on....

But just to be clear, let's look at SO(n,C) for n = 9 and n = 10, which illustrate the pattern even more clearly. I'll also explain how how it's all related to incidence geometry.

The Dynkin diagram of SO(9,C) has 4 = (9-1)/2 dots:

                 o-------o-------o====>====o 

To see how the dots correspond to different types of geometrical figures in some incidence geometry, first remember that we're starting with Cⁿ equipped with a symmetric bilinear form:

x⋅y = x₁ y₁ + ... + x_n y_n

This is really different than Rⁿ with its usual inner product, since it's perfectly possible for a vector to have x⋅x = 0, and we can even get big subspaces that are orthogonal to themselves. A subspace of Cⁿ is called "isotropic" if all vectors in this subspace are orthogonal to each other with respect to this form.

The idea of a subspace orthogonal to itself seems really weird at first! If you've never thought about this, you should probably skip ahead to the "addendum" at the end of this article, where I explain it in more detail. It's closely related to the fact that lightlike vectors in Minkowski spacetime are always orthogonal to themselves. In other words, they have x⋅x = 0.

To construct an incidence geometry for SO(n,C) and make it as similar to projective geometry as possible, we work not with Cⁿ but with the subspace of CP^n-1 coming from vectors in Cⁿ with x⋅x = 0. Algebraic geometers call this subspace a "quadric". In physics it arises naturally from taking (n-2)-dimensional Minkowski spacetime, compactifying it in a certain way, and then complexifying it - we'll talk about this more later! Inside this quadric there are various types of geometrical figures:

                 o-------o-------o====>====o 
            points     null     null      null
                      lines    planes    3-spaces

Putting together everything I've said so far: for n odd, the ith dot in the Dynkin diagram of Spin(n,C) corresponds to a maximal parabolic P such that Spin(n,C)/P is the manifold consisting of all isotropic i-dimensional subspaces in Cⁿ - or in other words, all null (i-1)-spaces in the corresponding quadric. And this manifold, called an "orthogonal Grassmannian", has a complex line bundle on it whose space of holomorphic sections is the ith fundamental rep of Spin(n,C).

For n even, let's look at SO(10,C).

The Dynkin diagram of SO(10,C) has 5 = 10/2 dots:

                                    o 
                                   /
                                  /
                                 /
                 o------o-------o 
                                 \
                                  \
                                   \
                                    o

In the language of incidence geometry, the dots again correspond to different types of figures in a quadric:

                                    o left-handed 4-spaces
                                   /
                                  /
                                 /
                 o------o-------o  null planes
              points   null      \
                      lines       \
                                   \
                                    o right-handed 4-spaces 

• points in the quadric, which are 1-dimensional isotropic subspaces of C¹⁰
• null lines in the quadric, which are 2-dimensional isotropic subspaces of C¹⁰
• null planes in the quadric, which are 3-dimensional isotropic subspaces of C¹⁰
• left-handed 4-spaces in the quadric, which are left-handed 5-dimensional subspaces of C¹⁰
• right-handed 4-spaces in the quadric, which are right-handed 5-dimensional subspaces of C¹⁰

** = -1

This means that its eigenvalues are i and -i. Thus there are "self-dual" 5-forms with

*w = iw

and "anti-self-dual" ones with

*w = -iw,

which give two kinds of 5-dimensional subspaces of C¹⁰ that are their own orthogonal complement: the so-called "right-handed" and "left-handed" ones. There's nothing special about the number 10 here; any even number n will do, though we should leave out the factor of "i" in the above formulas when n is a multiple of 4, since then the square of the Hodge star operator on n/2-forms is 1 instead of -1.

Okay, that pretty much concludes my story for Spin(n,C). I could do some other examples, but we're probably both getting worn out; if you want, you can read about them in section 23.3 of this book:

5) William Fulton and Joe Harris, Representation Theory - a First Course, Springer Verlag, Berlin, 1991.

So instead, let me conclude with a few remarks about twistors. taken from here:

6) Robert J. Baston and Michael G. Eastwood, The Penrose Transform: its Interaction with Representation Theory, Clarendon Press, Oxford, 1989.

The field equations for massless particles like photons are conformally invariant. The group SO(2,4) acts as conformal transformations of 4d Minkowski spacetime. To be precise, we should admit that some of these are just partially defined, like conformal inversion:

x |→ x/x⋅x

However, they become everywhere defined if we switch to a slightly bigger space, the "conformal compactification" of Minkowski spacetime.

The great realization of Roger Penrose was that it's nice to go even further and complexify this conformal compactification, getting a 4-dimensional complex manifold M with a complex metric. Minkowski spacetime sits inside this wonderful space M just like the real line sits inside the Riemann sphere. A lot of physics becomes easier on M, just like a lot of math is easier to do on the Riemann sphere than on the real line.

Now, since SO(2,4) is a real form of SO(6,C), the whole group SO(6,C) acts as symmetries of M. Of course the double cover Spin(6,C) also acts on M, so let's use that. Here's the cool part:

M = Spin(6,C)/P

where P is the maximal parabolic corresponding to this dot on the Dynkin diagram for Spin(6,C):

                          o
                         /
                        /
                       /
     we live here →  o
                       \
                        \
                         \
                          o

                          o left-handed planes
                         /
                        /
                       /
               points o 
                       \
                        \
                         \
                          o right-handed planes

If you unravel some of the definitions, this says that

M = {1-dimensional isotropic subspaces of C⁶},

so in physics lingo, M is the space of lightlike lines through the origin in C⁶.... but remember, these are complex lines.

So far, this stuff actually works in any dimension: the space of 1-dimensional isotropic subspaces of Cⁿ is the same as what you get by complexifying the conformal compactification of (n-2)-dimensional Minkowski spacetime, and so on.

But now we can use one of those charming coincidences:

Spin(6,C) = SL(4,C)!

This means we can also write

M = SL(4,C)/P

where now we think of P as a parabolic in SL(4,C). Let's see what M looks like in these terms. SL(4,C) acts on CP³, and we've seen that the dots in the Dynkin diagram for SL(4,C) correspond to these different types of geometrical figures in CP³:

                          o points
                         /
                        /
                       /
                lines o 
                       \
                        \
                         \
                          o planes

M = {lines in CP³}

or if you prefer:

M = {2-dimensional subspaces of C⁴}

Whew! What's the point? Well, these descriptions of the complexification of conformally compactified Minkowski spacetime let Penrose use incidence geometry methods to solve conformally invariant field equations, like Maxwell's equations or the Yang-Mills equations.

But what's a twistor? That's easy: it's just a spinor for Spin(6), either left-handed or right-handed. In other words, twistors are the fundamental reps corresponding to these dots on the Dynkin diagram:

                          o <--- left-handed twistors live here
                         /
                        /
                       /
                      o
                       \
                        \
                         \
                          o <--- right-handed twistors live here

There's a lot more to say, but I'll stop here! If you want more, try this:

7) S. A. Huggett and K.P. Tod, An Introduction to Twistor Theory, Cambridge U. Press, Cambridge, 1994.

Addendum: Someone who prefers to remain anonymous asked me to give some examples of "isotropic" subspaces of Cⁿ. I really should have done this earlier, because isotropic subspaces seem very mysterious before you've seen them, but very simple afterwards. They have a beautiful connection with special relativity, especially the geometry of light.

So, let me give some examples. But since complex numbers are weird, let's start with Rⁿ equipped with a metric of some signature or other, and look at the isotropic subspaces in there. An isotropic subspace is just a vector subspace where all vectors are orthogonal to each other. This is the same as a subspace in which all vector have xċx = 0 - or in physics lingo, one where all vectors are lightlike.

For starters consider good old Minkowski space, R^3,1. This has 3 space directions and 1 time direction, and it has a bunch of 1-dimensional isotropic subspaces. Why? Simple: these are just light rays through the origin.

Are there any 2-dimensional isotropic subspaces in Minkowski spacetime? No! To find one of these, we'd need two light rays through the origin that were orthogonal to each other. And this is impossible, basically because all lightlike vectors have a nonzero time component. To find two orthogonal light rays, we'd need to have two different time directions!

So, in R^3,1 the biggest isotropic subspaces are 1-dimensional. But if we had a spacetime like R^2,2, with two space directions and two time directions, we could find 2-dimensional isotropic subspaces. For example, if the metric on R^2,2 looks like this:

(x,y,s,t).(x',y',s',t') = xx' + yy' - ss' - tt'

then here are two lightlike vectors that are orthogonal to each other:

(1,0,1,0)

and

(0,1,0,1).

Since they are orthogonal, every linear combination of them is lightlike as well. So, these vectors span a 2d isotropic subspace.

Hopefully you get the picture now: to get an n-dimensional isotropic subspace in R^p,q we need at least n time dimensions and at least n space dimensions. So, there will be isotropic subspaces of dimensions going from zero on up to the minimum of p and q.

Now we're ready to bring the complex numbers into the story! We can take a real vector space with a metric on it and "complexify" it by letting our vectors have complex coefficients instead of real ones, and using the same formula for the metric. But the funny thing about "complexifying" is that it actually simplifies things in certain ways. Since i² = -1, you can turn a vector from timelike to lightlike or vice versa just by multiplying it by i! This means the distinction between space and time isn't such a big deal anymore. In particular, it doesn't matter how many space or time directions we had to begin with; after complexifying them, all the spaces R^p,q look just like Cⁿ (n = p+q) with the metric

x.y = x₁ y₁ + ... + x_n y_n

In other words, all these spaces R^p,q are sitting inside Cⁿ as different "real parts".

It's also easy to see that if we start with an isotropic subspace of R^p,q, and take complex linear combinations of the vectors in that subspace, we get an isotropic subspace of Cⁿ. This means all the stuff we just learned about the "real world" has ramifications for the "complex world".

For example, we instantly know that Cⁿ has isotropic subspaces of dimension up to the minimum of p and q, where p and q are any numbers with p+q = n. To get this minimum as big as possible, we should take p = q = n/2. Then we'll get isotropic subspaces of dimensions going all the way up to n/2. But we can only do this when n is even! When n is odd, the best we can do is (n-1)/2.

This shows that isotropic subspaces of Cⁿ work differently depending on whether n is odd or even. I described this in more detail above, where I separately treated SO(n,C) for n odd and n even.

Addendum: Here are two posts on sci.physics.research which address this mysterious fact: there's no dot in the Dynkin diagram for SO(2n,C) corresponding to the (n/2 - 1)-dimensional isotropic subspaces of C²ⁿ, even though there is one for every other dimension from 1 to n/2.

From: James Dolan
Subject: Re: This Week's Finds in Mathematical Physics (Week 180)
Date: Thu, 13 Jun 2002 


marc bellon wrote:

| John Baez writes:

|> Borcis wrote:

|> >John Baez wrote:
 
|> >>Boris Borcic wrote:
 
|> >> >None of the diagrams for C^n, n even, shows an entry for n/2-1 
|> >> >dimensional isotropic subspaces - how should we read this fact ?
 
|> >> I don't know what it means.  Isotropic subspaces of this
|> >> dimension certainly exist, but for some reason the theory
|> >> I am describing here does not regard them as important.  It's
|> >> not an arbitrary decision on anyone's part; it's built into the 
|> >> logic of the subject - but I don't understand it.
 
|> >No doubt a temporary phenomenon :)

|> Let's hope so.  

|Let me propose an explanation.
|It is sufficient to consider the four dimensional case.
|In the two-dimensional case, there are two isotropic lines,
|one of which is self-dual and the other anti-self-dual, so that 
|the configuration is completely fixed, consistent with the abelian 
|character of SO(2).
|Now when I choose an isotropic line is C^4, its orthogonal is a three
|dimensional subspace which contains it, so that the extension of the
|isotropic line to an isotropic plane is equivalent to choosing an 
|isotropic line in a two-dimensional space.  But in view of the 
|two-dimensional case, no choice has to be made, so that an isotropic
|line uniquely define two isotropic plane, one self-dual, the other
|anti-self-dual.  Reciprocally, a self-dual isotropic plane and an 
|anti-self-dual one evidently cannot coincide, but they cannot either 
|be complementary: in this case they are dual to each other using the 
|metric and are of the same self-duality.
|
|To give an isotropic line in C^4 is therefore equivalent to give a
|pair of isotropic planes, one self-dual and the other anti-self-dual.
|AFAIK, it is the property used in the twistor program of Penrose:
|you parameterize the light rays (null lines) by the isotropic planes
|it lies on.  More generally, when considering SO(2n), you do not need 
|to consider the (n-1)-dimensional isotropic plane, since they are 
|uniquely defined by the combination of a self-dual n-space and an
|anti-self dual one, if they have a (n-2)-dimensional space in common.


this seems like a good explanation.  extrapolating from this case,
maybe whenever we have a dynkin diagram corresponding to a particular
sort of incidence geometry, and a chosen dot in the diagram
corresponding to a particular sort of "point" in the geometry, then
for any "anti-chain" in the dynkin diagram, the type of partial flag
corresponding to the anti-chain is uniquely determined by (and thus
representable as) the intersection of the subspaces in the partial
flag.

thus in the case described by marc bellon, the dynkin diagram is a
"d" series diagram such as d_5:

      *
     /
*-*-*
     \
      *

and the chosen dot (actually an asterisk in the above picture) is the
leftmost one.  labeling the dots by letters and placing the chosen dot
at top we have:

  a
  |
 b
  |
  c
 / \
d   e

.  then a so-called "anti-chain" with respect to the partial order "x
is in more direct proximity to a than y is" is a dot-set s such that
no member of s is subordinate to any other member; thus for example
{}, {b}, and {d,e} are anti-chains but since e is subordinate to c,
{c,e} isn't an anti-chain.

{d,e} is in fact the only anti-chain in the above partial order with
more than one dot.  arranging the anti-chains in order from those with
a larger collection of subordinates to those with a smaller
collection, we have:

  {a}
   |
 {b}
   |
 {c}
   |
 {d,e}
  / \
{d} {e}
  \ /
  { }

.  now for each anti-chain we can try to calculate the dimension of
the intersection of all of the subspaces in a partial flag of the type
corresponding to the anti-chain (that is, containing one subspace of
each type corresponding to a dot in the anti-chain).  according to
marc bellon we get these dimensions:

  0
  |
 1
  |
 2
  |
  3
 / \
4   4
 \ /
  8

and this more or less explains the mystery which boris borcic and john
baez were discussing, as to why it seemed at first that 3-dimensional
subspaces play no interesting role in the incidence geometry of the
d_5 dynkin diagram (and correspondingly for other "d" series
diagrams): it turns out that 3-dimensional subspaces _do_ play an
interesting role here, but they're related to a multi-dot anti-chain
in the dynkin diagram instead of to a single dot.  the importance of
anti-chains here comes as a bit of a surprise if your intuition about
incidence geometry is based on classical projective geometry, where
the dynkin diagram is in the "a" series and the chosen dot is an
end-dot, because in that case there are no multi-dot anti-chains.

now we can take an arbitrary dynkin diagram and an arbitrary chosen
dot in it and try to calculate for the corresponding incidence
geometry the dimensions of the types of subspaces corresponding to the
anti-chains in the partial order, making some optimistic assumptions.
consider for example the dynkin diagram e_7:

    *
    |
*-*-*-*-*-*

with the rightmost dot as the chosen dot.  then we have:

  a
  |
 b
  |
 c
  |
  d
 / \
e   f
    |
    g

and the anti-chains for the partial order are:

  {a}
   |
 {b}
   |
 {c}
   |
 {d}
   |
 {e,f}
  / \
{f}{e,g}
  \ / \
  {g} {e}
    \ /
    { }

.  using an optimistic method of calculation related to methods
mentioned by john baez in some previous posts in this thread but not
really explained there either, we obtain for the dimensions of the
corresponding types of subspace:

  0
  |
 1
  |
 2
  |
 3
  |
  4
 / \
5   5
 \ / \
 10   6
   \ /
   27

.  so that's what this calculation predicts: that e_7 geometry
involves a compact 27-dimensional manifold of "points", with types of
special subspaces of dimensions 1, 2, 3, 4, 6, and 10, plus two
different types of special subspaces of dimension 5.  the special
4-dimensional subspaces and one of the types of special 5-dimensional
subspaces are evidently of "anti-chain" type.  i'd be interested to
know whether e_7 geometry has ever been described along these lines,
or more generally whether special subspaces of the "anti-chain" type
have been studied or at least noticed, beyond the cases described by
marc bellon.

From: James Dolan
Subject: Re: This Week's Finds in Mathematical Physics (Week 180)
Date: Sat, 15 Jun 2002 


i wrote:

|now we can take an arbitrary dynkin diagram and an arbitrary chosen
|dot in it and try to calculate for the corresponding incidence
|geometry the dimensions of the types of subspaces corresponding to the
|anti-chains in the partial order, making some optimistic assumptions.
|consider for example the dynkin diagram e_7:
|
|    *
|    |
|*-*-*-*-*-*
|
|with the rightmost dot as the chosen dot.  then we have:
|
|  a
|  |
 b
|  |
 c
|  |
|  d
| / \
|e   f
|    |
|    g
|
|and the anti-chains for the partial order are:
|
|  {a}
|   |
 {b}
|   |
 {c}
|   |
 {d}
|   |
| {e,f}
|  / \
|{f}{e,g}
|  \ / \
|  {g} {e}
|    \ /
|    { }
|
|.  using an optimistic method of calculation related to methods
|mentioned by john baez in some previous posts in this thread but not
|really explained there either, we obtain for the dimensions of the
|corresponding types of subspace:
|
|  0
|  |
 1
|  |
 2
|  |
 3
|  |
|  4
| / \
|5   5
| \ / \
| 10   6
|   \ /
|   27
|
|.  so that's what this calculation predicts: that e_7 geometry
|involves a compact 27-dimensional manifold of "points", with types of
|special subspaces of dimensions 1, 2, 3, 4, 6, and 10, plus two
|different types of special subspaces of dimension 5.  the special
|4-dimensional subspaces and one of the types of special 5-dimensional
|subspaces are evidently of "anti-chain" type.


having thought about it some more, i now think that we can give much
more specific information about the nature of the geometry here, and
in a much simpler way.


given a dotted dynkin diagram, this time for example say:

    a
    |
b-c-d-*-f-g-h

, we can consider the partially ordered set of all connected
sub-diagrams including the chosen dot, in this case:



                              *




                          d-*  *-f


                   a
                   |
                   d-*  c-d-*  d-*-f  *-f-g 


           a             a
           |             |
         c-d-*  b-c-d-*  d-*-f  c-d-*-f  d-*-f-g  *-f-g-h


            a      a
            |      |
        b-c-d-*  c-d-*-f  b-c-d-*-f  c-d-*-f-g  d-*-f-g-h


      a        a                       a
      |        |                       |
  b-c-d-*-f  c-d-*-f-g  b-c-d-*-f-g  c-d-*-f-g-h  b-c-d-*-f-g-h


                 a          a
                 |          |
             b-c-d-*-f-g  c-d-*-f-g-h  b-c-d-*-f-g-h


                             a
                             |
                         b-c-d-*-f-g-h



.  then each sub-diagram in the partial order can be interpreted as a
type of special subspace of the space of points in the

    a
    |
b-c-d-*-f-g-h

geometry, with the partial order (not completely explicit in the above
picture) indicating the containment relationships between the
subspaces in a complete so-called "flag" configuration, including
subspaces generated by intersection from the "principal" subspaces in
the flag.  furthermore, intersection of sub-diagrams corresponds
perfectly to intersection of subspaces in the flag.

thus in this case the space of points of the geometry contains special
subspaces that look like projective lines (since

*

is the dotted dynkin diagram for projective line geometry), two kinds
of special subspaces that look like projective planes (since

d-*

and

*-f

are slightly different ways of drawing the dotted dynkin diagram for
projective plane geometry), three kinds of subspaces that look like 
projective 3-spaces (since

a
|
d-*

and

c-d-*

and

*-f-g

are isomorphic to the dotted dynkin diagram for projective 3-space
geometry), and so forth.  since the

d-*

sub-diagram and the

*-f

sub-diagrams intersect in

*

, the intersection of special projective planes of the two different
types will be a special projective line if the two special projective
planes lie in a single flag.  and so forth.

(one minor defect in this treatment is that the semi-lattice of
connected sub-diagrams containing the chosen dot needs to be
supplemented by one extra element at the top to account for the
singleton subspaces of the geometry; i'm not going to worry about that
for now.)

now let's return to the example discussed by marc bellon.  we have a
d-series dynkin diagram dotted at the boring end, thus for example
d_5:

      d
     /
*-b-c
     \
      e

.  the semi-lattice of connected sub-diagrams containing the chosen
dot is:


           *


          *-b


         *-b-c


      d         *-b-c
     /               \ 
*-b-c                 e


              d
             /
        *-b-c
             \
              e


.  we see that a flag in this geometry includes a projective line
corresponding to

*

, a larger projective plane corresponding to

*-b

, a larger projective 3-space corresponding to

*-b-c

, two larger projective 4-spaces corresponding to

      d         *-b-c
     /               \ 
*-b-c                 e

whose intersection is the projective 3-space, and finally the space of
all points in the geometry, corresponding to


      d
     /
*-b-c
     \
      e


.  since the projective 3-space appears as the intersection of the two
projective 4-spaces, it's in some sense redundant and thus not one of
the "principal" subspaces in the flag.  but it's there nevertheless,
thus more or less resolving boris borcic's mystery of the missing
isotropic subspace.


this all seems simple enough (in principle) now that it must be
well-known, but i don't know where it might be discussed in reasonably
plain language.

© 2002 John Baez
baez@math.removethis.ucr.andthis.edu