
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 quarkgluon 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 Xrays reveal evidence for new form of matter, http://www1.msfc.nasa.gov/NEWSROOM/news/releases/2002/02082.html
The Chandra Xray 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: 2categorical 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 3space lying on...For other complex simple groups we'll get other concepts of "maximal flag", which I'll describe later.
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 5spaceor simply
a lineFor any parabolic subgroup P, the quotient space G/P is called a "flag manifold", since it's the space of all flags of the given type.
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:
ooo points lines planesSo, to pick out a flag manifold, you just mark the dots you want. For example,
xox points lines planesgives the parabolic P such that SL(4,C)/P is the space of all "points lying on a plane" in CP^{3}. As I explained earlier, this P is the subgroup of SL(4,C) consisting of all matrices of the form
* * * * 0 * * * 0 * * * 0 0 0 *If you look at the pictures in "week178", you should be able to figure out the recipe for getting this subgroup from a subset of dots in the Dynkin diagram, at least in the SL(n,C) case.
Even better, this game lets you get all the finitedimensional 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 finitedimensional 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:
ooo points lines planesThe irrep corresponding to this numbering:
100is the obvious representation of SL(n,C) on C^{n}. This irrep:
010is the obvious rep of SL(n,C) on the 2nd exterior power of C^{n}  or in physics lingo, rank two antisymmetric tensors. This irrep:
001is the obvious rep of SL(n,C) on the 3rd exterior power of C^{n}. And so on, if there are more dots. Note what we're really saying here: if you take the Grassmannian of all jdimensional subspaces in C^{n}, there's a godgiven complex line bundle on it whose space of holomorphic sections is the jth exterior power of C^{n}.
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 C^{n} preserving the symmetric bilinear form
x.y = x_{1} y_{1} + ... + 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:
ojust like the Dynkin diagram for SL(2,C). That's because they're isomorphic:
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 C^{2}. This fact is crucial for understanding spin1/2 particles in 3d space.
The Dynkin diagram of Spin(4,C) is two dots, not connected by an edge:
o ojust like the Dynkin diagram for SL(2,C) x SL(2,C). That's because they're isomorphic:
Spin(4,C) = SL(2,C) x SL(2,C).The fundamental reps coresponding to the two dots are called the "lefthanded" and "righthanded" spinor representations of Spin(4,C): they're just the obvious reps of SL(2,C) x SL(2,C) on C^{2}. This fact is crucial for understanding spin1/2 particles in 4d spacetime.
The Dynkin diagram of Spin(5,C) is two dots connected by a double edge:
o===>===oFor an explanation of the double edge and the arrow see "week62" and "week64", where I also explained that this Dynkin diagram is the same as that of Sp(4,C), the group of transformations preserving a symplectic structure on C^4. That's because these groups are isomorphic:
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 C^{5}  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 C^{4}  it's called the "spinor" rep of Spin(5,C). This would be fundamental for studying spin1/2 particles in 5dimensional spacetime if anyone were interested... but not many people are.
The Dynkin diagram of Spin(6,C) has three dots:
o / / / o \ \ \ oThis is the same as that of SL(4,C), though I've drawn it differently. That's because these groups are isomorphic:
Spin(6,C) = SL(4,C).
The fundamental rep corresponding to the left dot comes from the obvious rep of SO(6,C) on C^{6}  the "vector" rep again. The reps corresponding to the other dots are the left and righthanded spinor reps of Spin(6,C), coming from the obvious rep of SL(4,C) on C^{4} and its dual. This is fundamental for understanding spin1/2 particles in 6dimensional space  for example, the 6 extra curledup 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).They give rise to isomorphisms of their maximal compact subgroups, so let's say goodbye to those too:
Spin(3) = SU(2) Spin(4) = SU(2) x SU(2) Spin(5) = Sp(2) Spin(6) = SU(4).Sometime we should return and learn to know them better... they've barely begun to display their many charms! But today we must sail on to higher dimensions....
The Dynkin diagram of Spin(7,C) has three dots:
oo===>===oThe fundamental rep corresponding to the left dot comes from the vector rep of SO(7,C) on C^{7}. The rep corresponding to the middle dot is the second exterior power of the vector rep. The rep corresponding to the right dot is the spinor rep, which is no longer so easy to describe without using Clifford algebras  see "week93" or "week105" for more about those.
The Dynkin diagram of Spin(8,C) has four dots:
o / / / oo \ \ \ oThe fundamental rep corresponding to the left dot comes from the vector rep of SO(8,C) on C^{8}. The middle dot corresponds to the second exterior power of the vector rep. The top and bottom dots correspond to the left and righthanded spinor reps. Like the vector rep, these are also 8dimensional. This coincidence arises from the symmetry of the diagram, which is called "triality".
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 = (91)/2 dots:
ooo====>====oThe fundamental rep corresponding to the ith dot is the ith exterior power of vector rep, except for the last dot, which corresponds to the spinor rep.
To see how the dots correspond to different types of geometrical figures in some incidence geometry, first remember that we're starting with C^{n} equipped with a symmetric bilinear form:
x⋅y = x_{1} y_{1} + ... + x_{n} y_{n}
This is really different than R^{n} 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 C^{n} 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 C^{n} but with the subspace of CP^{n1} coming from vectors in C^{n} with x⋅x = 0. Algebraic geometers call this subspace a "quadric". In physics it arises naturally from taking (n2)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:
ooo====>====o points null null null lines planes 3spacesA "point" in the quadric is really a 1dimensional isotropic subspace of C^{n}; a "null line" is a 2dimensional isotropic subspace, and so on. We can talk about a point lying on a line, or a line lying on a plane, and they mean the obvious things. This gives the incidence geometry associated to Spin(n,C).
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 idimensional subspaces in C^{n}  or in other words, all null (i1)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 / / / ooo \ \ \ oThe fundamental rep corresponding to the ith dot is the ith exterior power of the vector rep, except for the last two dots, which correspond to the left and righthanded spinor reps.
In the language of incidence geometry, the dots again correspond to different types of figures in a quadric:
o lefthanded 4spaces / / / ooo null planes points null \ lines \ \ o righthanded 4spacesThe big difference from the odddimensional case is that there are two kinds of spaces of the highest dimension listed, and we leave out the nexthighest dimension. In our example we get:
** = 1
This means that its eigenvalues are i and i. Thus there are "selfdual" 5forms with
*w = iw
and "antiselfdual" ones with
*w = iw,
which give two kinds of 5dimensional subspaces of C^{10} that are their own orthogonal complement: the socalled "righthanded" and "lefthanded" 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/2forms 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 4dimensional 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 \ \ \ oWe've seen this diagram before. In the language of incidence geometry, the dots correspond to different figures in a quadric:
o lefthanded planes / / / points o \ \ \ o righthanded planesso points of M are just points of this quadric!
If you unravel some of the definitions, this says that
M = {1dimensional isotropic subspaces of C^{6}},
so in physics lingo, M is the space of lightlike lines through the origin in C^{6}.... but remember, these are complex lines.
So far, this stuff actually works in any dimension: the space of 1dimensional isotropic subspaces of C^{n} is the same as what you get by complexifying the conformal compactification of (n2)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 CP^{3}, and we've seen that the dots in the Dynkin diagram for SL(4,C) correspond to these different types of geometrical figures in CP^{3}:
o points / / / lines o \ \ \ o planesSo, we get yet another description of our marvelous spacetime:
M = {lines in CP^{3}}
or if you prefer:
M = {2dimensional subspaces of C^{4}}
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 YangMills equations.
But what's a twistor? That's easy: it's just a spinor for Spin(6), either lefthanded or righthanded. In other words, twistors are the fundamental reps corresponding to these dots on the Dynkin diagram:
o < lefthanded twistors live here / / / o \ \ \ o < righthanded twistors live hereIn the language of incidence geometry, these dots correspond to the two sorts of null planes in M. Penrose likes to think of these null planes as more fundamental than points....
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 C^{n}. 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 R^{n} 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 1dimensional isotropic subspaces. Why? Simple: these are just light rays through the origin.
Are there any 2dimensional 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 1dimensional. But if we had a spacetime like R^{2,2}, with two space directions and two time directions, we could find 2dimensional 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 ndimensional 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 i^{2} = 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 C^{n} (n = p+q) with the metric
x.y = x_{1} y_{1} + ... + x_{n} y_{n}
In other words, all these spaces R^{p,q} are sitting inside C^{n} 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 C^{n}. This means all the stuff we just learned about the "real world" has ramifications for the "complex world".
For example, we instantly know that C^{n} 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 (n1)/2.
This shows that isotropic subspaces of C^{n} 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 C^{2n}, 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/21 > >> >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 twodimensional case, there are two isotropic lines, one of which is selfdual and the other antiselfdual, 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 twodimensional space. But in view of the twodimensional case, no choice has to be made, so that an isotropic line uniquely define two isotropic plane, one selfdual, the other antiselfdual. Reciprocally, a selfdual isotropic plane and an antiselfdual 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 selfduality.  To give an isotropic line in C^4 is therefore equivalent to give a pair of isotropic planes, one selfdual and the other antiselfdual. 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 (n1)dimensional isotropic plane, since they are uniquely defined by the combination of a selfdual nspace and an antiself dual one, if they have a (n2)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 "antichain" in the dynkin diagram, the type of partial flag corresponding to the antichain 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 socalled "antichain" with respect to the partial order "x is in more direct proximity to a than y is" is a dotset s such that no member of s is subordinate to any other member; thus for example {}, {b}, and {d,e} are antichains but since e is subordinate to c, {c,e} isn't an antichain. {d,e} is in fact the only antichain in the above partial order with more than one dot. arranging the antichains 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 antichain 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 antichain (that is, containing one subspace of each type corresponding to a dot in the antichain). 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 3dimensional 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 3dimensional subspaces _do_ play an interesting role here, but they're related to a multidot antichain in the dynkin diagram instead of to a single dot. the importance of antichains 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 enddot, because in that case there are no multidot antichains. 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 antichains 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 antichains 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 27dimensional 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 4dimensional subspaces and one of the types of special 5dimensional subspaces are evidently of "antichain" 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 "antichain" 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 antichains 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 antichains 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 27dimensional 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 4dimensional subspaces and one of the types of special 5dimensional subspaces are evidently of "antichain" 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  bcd*fgh , we can consider the partially ordered set of all connected subdiagrams including the chosen dot, in this case: * d* *f a  d* cd* d*f *fg a a   cd* bcd* d*f cd*f d*fg *fgh a a   bcd* cd*f bcd*f cd*fg d*fgh a a a    bcd*f cd*fg bcd*fg cd*fgh bcd*fgh a a   bcd*fg cd*fgh bcd*fgh a  bcd*fgh . then each subdiagram in the partial order can be interpreted as a type of special subspace of the space of points in the a  bcd*fgh geometry, with the partial order (not completely explicit in the above picture) indicating the containment relationships between the subspaces in a complete socalled "flag" configuration, including subspaces generated by intersection from the "principal" subspaces in the flag. furthermore, intersection of subdiagrams 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 3spaces (since a  d* and cd* and *fg are isomorphic to the dotted dynkin diagram for projective 3space geometry), and so forth. since the d* subdiagram and the *f subdiagrams 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 semilattice of connected subdiagrams 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 dseries dynkin diagram dotted at the boring end, thus for example d_5: d / *bc \ e . the semilattice of connected subdiagrams containing the chosen dot is: * *b *bc d *bc / \ *bc e d / *bc \ 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 3space corresponding to *bc , two larger projective 4spaces corresponding to d *bc / \ *bc e whose intersection is the projective 3space, and finally the space of all points in the geometry, corresponding to d / *bc \ e . since the projective 3space appears as the intersection of the two projective 4spaces, 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 wellknown, but i don't know where it might be discussed in reasonably plain language.
© 2002 John Baez
baez@math.removethis.ucr.andthis.edu
