## This Week's Finds in Mathematical Physics (Week 186)

#### John Baez

Okay, now let's pull together all the strands of our story about Dynkin diagrams and q-mathematics. The story can be summarized in a rather elaborate diagram, of which this is the first part:

```                          DYNKIN DIAGRAM
/             \
/               \
pick a field /                 \
/                   \
/                     \
/        Weyl group     \
SIMPLE ALGEBRAIC ----------> COXETER GROUP
GROUP                        |
|                         |
FLAG VARIETY             COXETER COMPLEX
\                       /
\                     /
\                   /
\                 /
\               /
\             /
\           /
q-POLYNOMIAL
```
We start with a Dynkin diagram and see what we can do with it; we'll find that two separate routes lead to the same polynomial, which for lack of a better name I'll call the "q-polynomial". In recent weeks I've hinted that starting with the Dynkin diagrams in the An series, like this:
```                  o-------o-------o-------o-------o
```
we get the polynomials called "q-factorials". Now I'll sketch the story for arbitrary Dynkin diagrams!

Way back in "week62" I showed how a Dynkin diagram gives a finite reflection group: that is, a finite group of symmetries of n-dimensional Euclidean space, generated by reflections, one for each of the n dots in the diagram, satisfying relations described by the edges in the diagram. In fact, I noted that this trick works for a slightly more general class of diagrams called "Coxeter diagrams". The resulting groups are called "Coxeter groups".

But let's not go for maximal generality: any Dynkin diagram gives us a Coxeter group, and that's enough for now. Some of these Coxeter groups are symmetry groups of Platonic solids and their analogues in other dimensions: the regular polytopes. For example, starting with this Dynkin diagram:

```                           o-------o                  A2
```
we get the symmetry group of the equilateral triangle, while starting with this one:
```                           o===>===o                  B2
```
we get the symmetry group of the square, and starting with this one:
```                       o-------o--------o             A3
```
we get the symmetry group of the regular tetrahedron. Other Coxeter groups are symmetry groups of polytopes that aren't regular. This, for example:
```                                     o
|
|                E8
o----o----o----o----o----o-----o
```
is the symmetry group of a non-regular polytope in 8 dimensions with 240 vertices!

However, some Coxeter groups are not naturally regarded as the symmetry groups of polytopes. So, to deal with all Coxeter groups in a systematic way, it's better to think of them as symmetries of certain simplicial complexes called "Coxeter complexes". Roughly speaking, a simplicial complex is a gadget made of points, line segments, triangles, tetrahedra, 4-simplices, and so on - all stuck together in a nice way.

If you have a Coxeter diagram with n dots, the highest dimension of the simplices in its Coxeter complex will be n-1, and it will have one of these top-dimensional simplices for each element of the Coxeter group. For example, I've already said this Dynkin diagram:

```                           o-------o                   A2
```
gives the Coxeter group consisting of symmetries of the equilateral triangle - by which I mean all reflections and rotations. This group has 6 elements, so the Coxeter complex is built from 6 line segments together with lower-dimensional simplices (points) - and in fact, it's just a hexagon.

A hexagon is also what you get by dividing each edge of the equilateral triangle into two parts. That's no coincidence: whenever our Coxeter group is naturally the symmetries of a polytope, we can get the Coxeter complex by "barycentrically subdividing" the surface of this polytope - which basically means sticking an extra vertex in the middle of every face of the polytope and using these to chop its surface into simplices.

For example, this diagram

```                       o-------o--------o              A3
```
gives the symmetry group of the tetrahedron, so we can get its Coxeter complex by barycentrically subdividing the surface of the tetrahedron, obtaining a shape with 24 triangles. Surprise: this is just the size of the symmetry group of the tetrahedron!

But that's how it always works: the number of top-dimensional simplices in the Coxeter complex is the number of elements in the Coxeter group. Even better, if you pick any top-dimensional simplex in the Coxeter complex, there always exists a unique element of the Coxeter group that maps it to any other top-dimensional simplex. So the Coxeter complex is the best possible thing made out of simplices on which the Coxeter group acts as symmetries.

Now, all of this has been done starting with a Dynkin diagram and nothing else. But we can do other stuff if we pick a field, like the real numbers R or the complex numbers C - or if you're feeling daring, the field Fq with q elements, where q is some power of a prime number.

First and most importantly, a field lets us define a "simple algebraic group". If we use R or C as our field these are just the usual real or complex simple Lie groups associated with Dynkin diagrams, which I explained in "week63" and "week64". These are tremendously important in physics, and that's what got me going on this business in the first place! But we can also mimic this procedure using other fields, and if we use the finite field Fq, we get fascinating connections to q-mathematics... which I've begun explaining in recent Weeks.

No matter what field we use, the group we get will be the symmetries of a kind of "incidence geometry": a setup with stuff like points, lines, and planes, but perhaps also other geometrical figures that they never told you about in school. There will be one type of geometrical figure for each dot in our Dynkin diagram!

In the case where our field is the complex numbers, I explained these incidence geometries rather carefully in "week178", "week179" and "week180". But they're pretty similar for other fields, so to a zeroth approximation you can sort of fake it and pretend they work just the same. Eventually that attitude will get you in trouble, but hopefully you'll notice when it happens.

For example, the Dynkin diagram An has n dots in a row like this:

```                  o-------o-------o-------o-------o
```
and this gives the symmetry groups of projective geometry: the geometry of points, lines, planes, and so on up to dimension n.

More precisely, if we pick any field F, we can use this diagram to concoct the group SL(n+1,F) consisting of (n+1) x (n+1) matrices with entries in F and determinant equal to 1. This group acts on the projective n-space FPn - the space of all 1-dimensional subspaces of the vector space Fn+1. Just as in the complex case, we can talk about points, lines, planes and the like in FPn, and also incidence relations like "this point lies on that line". These relations satisfy the axioms of projective geometry, as explained in "week162". The group SL(n+1,F) acts on all these geometrical figures in a way that preserves the incidence relations... so we say it's a symmetry group for this particular projective geometry!

(If you prefer the group PSL(n+1,F), that's fine too; maybe even better. They have the same Lie algebra so it's not all that big a deal.)

The same general sort of thing works for all other Dynkin diagrams, too. The Bn and Dn series give the symmetry groups of conformal geometries, while the Cn series give the symmetry groups of symplectic geometries, and the exceptional Dynkin diagrams give symmetry groups of "exceptional geometries" associated to the octonions and their analogues for other fields.

In general, whenever we pick a Dynkin diagram and a field we get a geometry. We define a "maximal flag" in this geometry to consist of one geometrical figure of each type, all incident. The set of maximal flags turns out to be the key to understanding all the different kinds of incidence geometry in a unified way. When our field is the real or complex numbers this set is a manifold, often called the "flag manifold" - it's a special case of the flag manifolds described in "week180". But over other fields, the set of maximal flags is not a manifold but an "algebraic variety". If you don't know what that means, don't worry: I'm only mentioning this because then we get to call it the "flag variety" and sound intelligent. The real point here is that there's a wonderful analogy:

```       SIMPLE ALGEBRAIC GROUPS    |      COXETER GROUPS
-----------------------------|----------------------
FLAG VARIETIES         |    COXETER COMPLEXES
```
Just as a Coxeter group acts as symmetries of its Coxeter complex, a simple algebraic group acts as symmetries of its flag variety. But the analogy goes far deeper than that! In a certain strange way, you really can think of the Coxeter group as a simple algebraic group over the field Fq where q = 1, and you can think of the Coxeter complex as the corresponding flag variety.

Of course, there is no field Fq with q = 1. Nonetheless, all sorts of formulas that work for other values of q for simple algebraic groups over Fq and their flag varieties, apply when q = 1 to Coxeter groups and their Coxeter complexes! I gave the primordial example in "week184", which comes from the Dynkin diagram An. The number of points in the flag variety of the group SL(n+1,Fq) is the q-factorial

```[n+1]! =   ... [n+1]
```
where
```[i] = 1 + q + q2 + ... + qi-1
```
When we set q = 1 in this formula, we get the ordinary factorial (n+1)!, and this is the number of total orderings of an n-element set. It's also the number of top-dimensional simplices in the Coxeter complex for An - and that's the way to think about it that works for other Dynkin diagrams.

In general, the trick is to set up a kind of incidence geometry starting from the Coxeter complex, in which the top-dimensional simplices serve as maximal flags, and the 0-simplices serve as geometrical figures of the various types... where two figures are "incident" if the 0-simplices are both vertices of some top-dimensional simplex!

To get a tiny taste of how this stuff works, consider the Dynkin diagram A2. We've seen that the Coxeter complex is a barycentrically subdivided triangle:

```                               x
/ \
/   \
/     \
/       \
o         o
/           \
/             \
/               \
/                 \
x---------o---------x
```
or viewed a bit differently, a hexagon:
```                            x-----o
/       \
/         \
o           x
\         /
\       /
x-----o
```
Here the vertices marked x are the vertices of the original triangle, while the vertices marked o correspond to its edges. We make up a puny little geometry where the x's are called "points" and the o's are called "lines". And we say a point and a line are "incident" if the x and o are the two ends of a line segment.

Note that any two distinct points are incident to a unique line, and any two distinct lines are incident to a unique point! This is characteristic of projective plane geometry. And that's just right, because A2 is the Dynkin diagram corresponding to projective plane geometry. If we do projective plane geometry over a field F, the group SL(3,F) acts as symmetries. But for this puny little geometry, the Coxeter group acts as symmetries. This is the symmetry group of the triangle, which is the group of permutations of its three vertices.

More generally, suppose we start with the diagram An. Then we'd see that its Coxeter group consists of permutations of n+1 things: the vertices of an n-simplex. The Coxeter complex would be gotten by barycentrically subdividing the surface of this n-simplex. And the Coxeter group would act on a puny little geometry built from the Coxeter complex, very much as SL(n+1,F) acts on the projective space FPn.

As I explained in "week184" and "week185", this relation between permutation groups and the groups SL(n+1,F) is just the tip of a very big iceberg. What I'm saying now is that a similar story works for all the other Dynkin diagrams, too!

To explain how this works, I'd need to tell you about the "Bruhat decomposition" of a flag variety. And to explain it really well, I'd need to tell you about Jacques Tits' theory of "buildings". Jim Dolan and I have been studying this over the last year, and it's really cool... but alas, it's too big a subject to explain here! So think of this Week as a mere advertisement for the theory of buildings, if you like. I'll give you some references at the end.

Okay. So far I've talked about two kinds of things we can get from Dynkin diagrams: "flag varieties", if we pick a field, and "Coxeter complexes", where we don't need to pick a field. Now let's bring in the q-mathematics! It turns out that that we can decategorify either the flag variety or the Coxeter complex and get something I call the "q-polynomial".

We can define this polynomial in four equivalent ways:

a) the coefficient of qi in this polynomial is the number of Coxeter group elements of length i. Here we "length" of any element in the Coxeter group is its length as a word when we write it as product of the generating reflections.

b) the coefficient of qi in this polynomial is the number of top-dimensional simplices of distance i from a chosen top-dimensional simplex in the Coxeter complex. Here we measure "distance" between top-dimensional simplices in the hopefully obvious way, based on how many walls you need to cross to get from one to the other.

c) the coefficient of qi in this polynomial is the number of i-cells in the Bruhat decomposition of the flag variety. Here the "Bruhat decomposition" is a standard way of writing the flag variety as a disjoint union of "i-cells", that is, copies of Fi where F is our field and i is a natural number. These i-cells are called either "Bruhat" or "Schubert" cells, depending on who you talk to.

d) the coefficient of qi is the rank of the (2i)th homology group of the flag variety defined over the complex numbers. More precisely: this homology group is isomorphic to Zk for some natural number k, called the "rank" of the homology group.

It's easy to see that a) and b) are equivalent; ditto for c) and d). The equivalence between b) and c) is deeper; it comes from the wonderful analogy between Coxeter complexes and flag varieties.

Let's calculate the q-polynomial of A2 using method b):

```                              0
x-----o
1 /       \ 1
/         \
o           x
\         /
2 \       / 2
x-----o
3
```
I've written down the distance of each top-dimensional simplex from a given one. There's one of distance 0, two of distance 1, two of distance 2, and 1 of distance 3. This gives
```q3 + 2q2 + 2q + 1 =  !
```
just as it should.

We can distill all sorts of nice information from the q-polynomial. For example, starting from facts a) - d) we immediately get:

e) the degree of this polynomial is the maximum length of an element of the Coxeter group. There is in fact a unique element with maximum length, called the "long word".

f) the degree of this polynomial is the dimension of the flag variety over any field.

and also:

g) the value of this polynomial at q a prime power is the cardinality of the flag variety over the field Fq.

h) the value of this polynomial at q = 1 is the number of elements in the Coxeter group.

i) the value of this polynomial at q = 1 is the Euler characteristic of the flag variety over the complex numbers.

j) the value of this polynomial at q = -1 is the Euler characteristic of the flag variety defined over the real numbers.

We can summarize this network of relations in the following diagram:

```                          DYNKIN DIAGRAM
/             \
/               \
pick a field /                 \
/                   \
/                     \
/       Weyl group      \
SIMPLE ALGEBRAIC ----------> COXETER GROUP
GROUP                        |
|                         |
FLAG VARIETY             COXETER COMPLEX
\                       /
\                     /
\                   /
\                 /
\               /
\             /
\           /
q-POLYNOMIAL
value at a prime power q /   |  |  |  \degree
/    |  |  |   \
number of points in /     |  |  |  dimension of flag variety =
flag variety over F_q       |  |  |  length of longest word in Coxeter group
|  |  |
|  |  |
value at q = 1|  |  |ith coefficient
|  |  |
number of Coxeter group | number of Coxeter group
elements | elements of length i =
= number of cells | number of i-cells
in flag variety | in flag variety =
= Euler characteristic of | rank of (2i)th homology group of
flag variety over C | flag variety over C
|
|
|value at q = -1
|
Euler characteristic of flag variety over R
```
Besides things I've already explained, I stuck in an extra arrow showing that you can get the Coxeter group from a simple algebraic group by forming something called its "Weyl group". I explained this connection way back in "week62". If we work over the real numbers and use the compact real form of our simple Lie group, the Weyl group acts on the Lie algebra of the maximal torus of this group - the so-called "Cartan algebra". In this context it's good to think of the Coxeter complex as sitting inside the Cartan algebra!

Next week I'll go through a bunch of examples. Right now, let me just give you some references for further reading.

To understand most of what I'm saying you mainly just need to understand the "Bruhat decomposition" of the flag variety. For a quick sketch of how this works over the complex numbers, try this book:

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

For a treatment of it over arbitrary fields, try:

2) Francois Digne and Jean Michel, Representations of Finite Groups of Lie Type, London Mathematical Society Student Texts 21, Cambridge U. Press, Cambridge, 1991.

But to understand the relation to incidence geometry, it will help a lot if you eventually study "buildings". This subject has a certain reputation for obscurity. One good place to start is this book written by someone who was himself trying to understand the subject:

3) Kenneth S. Brown, Buildings, Springer, Berlin, 1989.

Another is this:

4) Paul Garrett, Buildings and Classical Groups, Chapman & Hall, London, 1997.

For a lot more information about how finite simple groups show up as symmetries of buildings, try:

5) Antonio Pasini, Diagram Geometries, Oxford U. Press, Oxford, 1994.

and for the original source, go to:

6) Jacques Tits, Buildings of Spherical Type and Finite BN-pairs, Springer Lecture Notes in Mathematics 386, Berlin, New York, 1974.

Even better, come and sit in on Jim Dolan's seminar on the subject, here at UCR!