The n-Category Café

From what I understand, they are calculating the representations of the split real form of E8 — i.e. the real Lie group with E8 Dynkin diagram and which has a maximal torus which is a product of R+’s (rather than some R+ and some S¹ — e.g. SL_n(R) has a torus, the diagonal matrices or their connected component, which is a product of R+’s, but SU_n does not — this uniquely characterizes the real form).

The Kazhdan–Lusztig polynomials have two interpretations, identified by the Kazhdan–Lusztig conjectures (Beilinson–Bernstein, Vogan–Lusztig in the real case) — one is by counting points like you explained, the other as the multiplicities of irreducible representaitons in “standard” (basically induced) representations. So one way to say what they did was calculate the multiplicities of simples in standards for the split real form of E8. (Well, the web page said they did a block of this calculation — the matrix of multiplicities has block diagonal form — but maybe it’s known how to reduce everything to this block).

Geometrically this can also be said in terms of counting points, but now for orbits on the flag variety of E8 not of upper triangular matrices, but of a group K_C = complexification of the maximal compact subgroup of the real group E8(R) – such subgroups always have finitely many orbits on the flag variety but their combinatorics is MUCH more involved than that of Schubert varieties.

Hope this helps somewhat…
David

This paper by Jeffrey Adams and Fokko du Cloux may help. See p. 46 for the 453060 figure.

Thank you for the interesting post! You’re right about the AIM having coaxed a lot of science reporters to write about it: there’s a short note in Le Monde today here, which is very unusual :-)

This story about $\mathrm{E}_8$ also made the BBC:

• BBC News, 248-dimensional maths puzzle solved.

It’s an accurate story without any silly comparisons to the human genome project.

Clearly the American Institute for Mathematics has a good PR team! If it gets more people interested in math, it’s probably a good thing.

Then, if we fix two Schubert cells, we can count the number of points in the intersection of their closures. This number will be some polynomial in $q$. This is called a Kazhdan–Lusztig polynomial.

This isn’t quite right. This is what is called the $R$ polynomial. The Kazhdan–Lusztig polynomials are related to the $R$ polynomials by a linear map, but the actual geometric meaning of the Kazhdan–Lusztig polynomials is a good deal more subtle.

To explain what Kazhdan–Lusztig polynomials mean, let me start with one of the most beautiful stories in 20th century mathematics, the proof of the Weil Conjectures. Suppose that $M$ is a compact, oriented manifold and $F$ is an automorphism. Then, by the Lefschetz trace formula, the number of fixed points of $F$ is

$$\sum_{i=0}^{dim M} (-1)^i Tr(F^*: H^i(M,Q) \to H^i(M,Q) )$$

Now, let’s suppose that $M$ is not a manifold anymore, but a smooth subvariety of projective space, defined by equations over the field $F_q$. Then the $F_q$ points of $M$ are precisely the points fixed by the Frobenius automorphism

$$F: (x_0 : x_1 : ... : x_n) \mapsto (x_0^q : x_1^q : ... : x_n^q)$$

of $M$. Grothendieck and Deligne were able to define a cohomology theory, called etale cohomology, which works for varieties over $F_q$ and for which the Lefschetz trace formula still holds. Using this theory, they were able to prove Weil’s conjectures, which are statements about how the number of $F_q$ points on M can vary with $q$.

If we want to use these ideas to count $F_q$ points on the intersection of two Schubert cells, one thing gets harder and one thing gets easier. The harder thing is that the intersection of two Schubert cells is (usually) singular. So one needs to develop a version of etale cohomology for singular spaces. The relevant theory is called “intersection cohomology”, and it is a new and very confusing idea even when we work with honest singular complex varieties. One of the aspects of the new theory is that the contribution of a single fixed point of the Lefschetz trace formula is not just 1, but another alternating sum of traces which measures how singular the space is at that fixed point. The thing that is nice though is that, once you get all of the definitions right, in this particular Schubert example the action of Frobenius on the intersection cohomology groups is just by powers of $q$. Thus, you can state everything in terms of polynomials in $q$, not traces of large matrices.

Let $X_v$ and $X_w$ be two Schubert cells, $X_v$ contained within $X_w$. Then KL polynomials describe the intersection cohomology of $X_w$ near a “typical” point of $X_v$. One way to compute this is to count points on intersections of Schubert varieties and then work backwards from the Lefshetz trace formula, although apparently that’s not what was actually done here.

Every thing I’ve written is equally correct (and, hopefully, also correct :)) for flag varieties or for the $E_8$ versions of flag varieties.

Here’s an email from Jim Dolan and my reply. Any clarification from experts would be helpful!

by the way, no big deal, but since i know that you know that the equivalence classes are called the “bruhat classes” and their closures are called the “schubert cells”, and you decided to talk about both of these, i’m wondering why you didn’t think it would be clearer to go ahead and use the standard names for them.

Well, the paper I was reading to understand this stuff:

• Francesco Brenti, Kazhdan-Lusztig polynomials: history, problems, and combinatorial invariance.

calls the open things “Schubert cells” and their closures “Schubert varieties”. Since I was referring to this paper, I didn’t want to use different terminology.

It suggests that we’re not the only ones who found this terminology a bit slippery.

Is there a file out there with the set of generators as permutations? Being a computer scientist I am more interested in the permutation structure than the geometrical one.

“The trick for getting compact Lie groups from the distance function on their maximal torus is something kids learn in grad school.”….describing them as kids made me read that as *grade* school…for a moment there I thought something was seriously wrong with my primary education.

This has a flavor vaguely similar to the $E_{10}$-genome project that H. Nicolai and collaborators are working on: they try to list the decomposition of $e_{10}$ into irreps, ordered by “level”, of the action of various of its sub-Kac-Moody algebras.

Of course I know that this is different from what the Atlas project is doing. On the other hand, the not-quite-as-wild-as-it-may-sound speculation is that the $E_{10}$-genome project is of importance comparable to that of the human genome project, in that it might actually say something about the physical world we live in.

And then, there is a closer relation: $e_{8}$ is a sub-algebra of $e_{10}$, too, and one may go ahead and decompose $e_{10}$ into irreps of $e_{8}$, i.e. into those reps that the Atlas project has listed. This might be interesting, for the reasons John and I talked about here.

Here’s another article about this $\mathrm{E}_8$ computation:

• Hazel Muir, Mathematicians finally map 248-dimensional structure, New Scientist.

This is exciting news. As the team computed Kazhdan-Lusztig-Vogan polynomials for the split real form of E8, this might help one better understand the symmetries of the 57 dimensional charge-entropy space of BPS black hole solutions.

M. Gunaydin, K. Koepsell and H. Nicolai investigated this 57D space in their 2001 paper:

Conformal and Quasiconformal Realizations of Exceptional Lie Groups.

M. Gunaydin, A. Neitzke, B. Pioline and A. Waldron give a more recent overview in:

BPS black holes, quantum attractor flows and automorphic forms.

If I recall correctly, the construction of the 57 dimensional space involves adding an extra real coordinate to the 56 dimensional Freudenthal triple system over the split-octonions.

Here is an excerpt of an email from Jeffrey Adams of the University of Maryland, College Park. He’s part of the team who did this $\mathrm{E}_8$ calculation. He kindly corrected some of the guesses I made in my feeble attempt at explaining their work.

In what follows $G$ is a complex semisimple Lie group and $B$ is a Borel subgroup.

The original Kazhdan–Lusztig polynomials were defined for the flag variety $B\backslash G/B$ ($G,B$ complex), and are related to representations of the complex Lie algebra $\mathbf{g}$ in category $O$, also known as Verma modules.

The Kazhdan–Lusztig–Vogan polynomials are defined for the variety $K\backslash G/B$, where $G,B$ are as before (complex), and $K$ is the complexified maximal compact subgroup of a real form $G(\mathbb{R})$ of $G$. These are related to admissible representations of $G(\mathbb{R})$. Due to Vogan’s modesty, he refers to these as Kazhdan–Lusztig polynomials. In the category $O$ case, a Verma module is attached to an orbit of $B$ on $G/B$. Note that the stabilizer of a point in $G/B$ is connected. In the $G(\mathbb{R}$) case, a representation is associated to a local system on an orbit $K$ on $G/B$. The stabilizer of a point may not be connected; the number of local systems is the component group. Hence there are more representations than orbits.

Here is an example using the atlas software:

sophus-t43:~% atlas
This is the Atlas of Reductive Lie Groups Software Package 
version 0.2.5.
Build date: Nov 24 2006 at 09:16:16.
Enter "help" if you need assistance.
empty: type
Lie type: C2 sc s
###This is Sp(4,C), type C_2, simply 
###connected, split inner class
main: kgb
(weak) real forms are:
0: sp(2)
1: sp(1,1)
2: sp(4,R)
enter your choice: 2
###Now I've specified the split group Sp(4,R)
kgbsize: 11
###That's the number of orbits of K 
###on G/B
Name an output file (hit return for stdout):
 0:     1   2     6   4  [nn]  0
 1:     0   3     6   5  [nn]  0
 2:     2   0     *   4  [cn]  0
 3:     3   1     *   5  [cn]  0
 4:     8   4     *   *  [Cr]  1  2
 5:     9   5     *   *  [Cr]  1  2
 6:     6   7     *   *  [rC]  1  1
 7:     7   6    10   *  [nC]  2  2,1,2
 8:     4   9     *  10  [Cn]  2  1,2,1
 9:     5   8     *  10  [Cn]  2  1,2,1
10:    10  10     *   *  [rr]  3  1,2,1,2
###More information about the 11 orbits 0,...,10
real: block
possible (weak) dual real forms are:
0: so(5)
1: so(4,1)
2: so(2,3)
enter your choice: 2
###Now I've specified a "block" of representations
###This is a certain set of representations
Name an output file (hit return for stdout):
 0( 0,6):   1   2    ( 6, *)  ( 4, *)    [i1,i1]  0
 1( 1,6):   0   3    ( 6, *)  ( 5, *)    [i1,i1]  0
 2( 2,6):   2   0    ( *, *)  ( 4, *)    [ic,i1]  0
 3( 3,6):   3   1    ( *, *)  ( 5, *)    [ic,i1]  0
 4( 4,4):   8   4    ( *, *)  ( *, *)    [C+,r1]  1  2
 5( 5,4):   9   5    ( *, *)  ( *, *)    [C+,r1]  1  2
 6( 6,5):   6   7    ( *, *)  ( *, *)    [r1,C+]  1  1
 7( 7,2):   7   6    (10,11)  ( *, *)    [i2,C-]  2  2,1,2
 8( 8,3):   4   9    ( *, *)  (10, *)    [C-,i1]  2  1,2,1
 9( 9,3):   5   8    ( *, *)  (10, *)    [C-,i1]  2  1,2,1
10(10,0):  11  10    ( *, *)  ( *, *)    [r2,r1]  3  1,2,1,2
11(10,1):  10  11    ( *, *)  ( *, *)    [r2,rn]  3  1,2,1,2
###This says there are 12 representations,
###0-11. In the pair (x,y), x is an orbit
###of K on G/B. All orbits are connected,
###except #10; representations 10 and 11 
###both live on orbit #10.

One very interesting thing happens if you apply the $G(\mathbb{R})$ to a complex group, viewed as a real group. Let $H$ be a complex group, such as $GL(n,\mathbb{C})$, viewed as a real group. Then $H(\mathbb{C})=H \times H$, $B(\mathbb{C})=B\times B$, $K(\mathbb{C})=H(C)^\Delta$ (diagonally embedded). Then

$$H(\mathbb{C})^\Delta \backslash H(\mathbb{C})\times H(\mathbb{C})/B(\mathbb{C})\times B(\mathbb{C}) = B(\mathbb{C})\backslash G(\mathbb{C})/B(\mathbb{C})$$

As a result the category $O$ and $G(\mathbb{R})$ theories coincide: the orbit spaces are the same. On the level of representations this is subtle: a relationship between representation in category $O$ and representations of a complex Lie group.

It isn’t so easy to give references for this stuff. My paper with Fokko du Cloux, on turning representation into an algorithm for a computer, has lots of information. This distills down all of the machinery of real representation theory into a finite algorithm, which is what the atlas software implements. See:

• Jeffrey Adams and Fokko du Cloux, Algorithms for representation theory of real reductive groups.

I hope this is helpful. I’ll stay tuned to your blog.

Jeff

Here’s another article about this $\mathrm{E}_8$ stuff, featuring a picture of Jeffrey Adams drawing something very symmetrical:

• Kenneth Chang, The scientific promise of perfect symmetry, New York Times, March 20, 2007.

Hello.
I read your article very helpful.
Thank you.
So I introduce your article in my blog.
Not translate, just link your article.

‘Most beautiful’ math structure appears in lab for first time
07 January 2010 by David Shiga

“A complex form of mathematical symmetry linked to string theory has been glimpsed in the real world for the first time, in laboratory experiments on exotic crystals.”

“Mathematicians discovered a complex 248-dimensional symmetry called E8 in the late 1800s….”

“Radu Coldea of the University of Oxford and his colleagues chilled a crystal made of cobalt and niobium to 0.04 °C above absolute zero. Atoms in the crystal are arranged in long, parallel chains. Because of a quantum property called spin, electrons attached to the atom chains act like tiny bar magnets, each of which can only point up or down.”

“Strange things occurred when the experimenters applied a powerful 5.5-Tesla magnetic field perpendicular to the direction of these electron ‘magnets.’ Patterns appeared spontaneously in the electron spins in the chains – in a simplified example with three electrons, the spins could read up-up-down or down-up-down, among other possibilities. Each distinct pattern has a different energy associated with it.”

“The ratio of these different energy levels showed that the electron spins were ordering themselves according to mathematical relationships in E8 symmetry.”

“Alexander Zamolodchikov, currently at Rutgers University in Piscataway, New Jersey, pointed out in 1989 that the theoretically predicted energies of such systems match expectations from E8 symmetry….”

Journal Reference:
Quantum Criticality in an Ising Chain: Experimental Evidence for Emergent E8…

Coldea et al.
Science 8 January 2010: 177-180
DOI: 10.1126/science.1180085

The quickest way to describe E8 may be this. Take equal-sized balls in 8 dimensions and find the densest possible lattice packing.

Is this 7-dimensional solid balls inside 8-D? Or equal-sized balls of dimension 8? Or are we talking about regular 3-D balls packed in 8-D?