## March 19, 2007

### News about E8

#### Posted by John Baez

The exceptional Lie group E8 is a marvelous 248-dimensional monster, with mysterious connections to the octonions and string theory. Here’s a nice webpage about a new calculation involving $\mathrm{E}_8$:

As part of a project called the Atlas of Lie Groups and Representations, a team of mathematicians led by Jeffrey Adams have computed the Kazhdan–Lusztig–Vogan polynomials for $\mathrm{E}_8$.

You may have heard some hype about this, because it’s a really big calculation, and the American Institute of Mathematics has coaxed a lot of science reporters to write about it — in part by comparing it to the human genome project.

To see what was really done, try these:

Computing the Kazhdan–Lusztig–Vogan polynomials for $\mathrm{E}_8$ is certainly nowhere nearly as important as the human genome project, nor as hard!

However, the final result involves more data, in a sense. The answer is a 453,060 × 453,060 matrix of polynomials which takes 60 gigabytes to store. For comparison, see my webpage on information. The human genome is a mere 1 gigabyte – a pickup truck full of books. A good recording of the complete works of Beethoven takes 20 gigabytes. A library floor of academic journals holds 100 gigabytes.

So, the computation was indeed big.

But what’s $\mathrm{E}_8$, and what’s a Kazhdan–Lusztig–Vogan polynomial? You probably won’t hear the popular media tackle those questions. Let me give it a try.

The quickest way to describe $\mathrm{E}_8$ may be this. Take equal-sized balls in 8 dimensions and find the densest possible lattice packing. If you center one ball at the origin, the centers of the balls form a lattice $L \subset \mathbb{R}^8,$ that is, a discrete set closed under addition and subtraction.

Then, form the quotient $\mathbb{R}^8/L$ which is an 8-dimensional torus. There’s a way to measure distances on this torus, coming from the usual way to do this in $\mathbb{R}^8$. We’ve just wrapped up 8-dimensional Euclidean space into a torus!

Next, it’s a fact that every compact Lie group has a ‘maximal’ torus — a subgroup shaped like a torus that’s as big as possible while having this property. In fact there are lots of maximal tori, but they’re all alike, so people usually speak of ‘the’ maximal torus. You can measure distances on the maximal torus, since there’s a unique distance function on the Lie group that’s invariant under all the symmetries the group has. (Unique up to an overall scale factor, anyway — let’s not worry about that.)

The cool part is this: if someone hands you the maximal torus of a compact Lie group, together with how to measure distances on this torus, you can recover the Lie group! At least, you can if you’ve taken a course on this stuff.

So, the 8-dimensional torus I just described comes from a unique compact Lie group, and this group is $\mathrm{E}_8$!

I said this might be the quickest way to describe $\mathrm{E}_8$. But, to actually use this description to get your hands on $\mathrm{E}_8$ is not so easy! The trick for getting compact Lie groups from the distance function on their maximal torus is something kids learn in grad school. The really hard part is to find the densest lattice packing of balls in 8 dimensions. Luckily, you can just look it up. This lattice consists of all vectors $(x_1,\dots,x_8)$ such that

• the numbers $x_i$ are either all integers or all half-integers (a ‘half-integer’ being an integer plus 1/2)
and
• the sum $x_1 + \cdots + x_8$ is even.

Here’s a picture of the densest lattice packing of balls in 8 dimensions:

Drawn by John Stembridge, this shows the centers of all 240 balls that touch a given one, with the center of each ball connected by a line to the centers of its nearest neighbors. Of course, the picture has been projected from 8 dimensions down to a mere 2.

Not very easy to visualize! If you’re really interested in this stuff, you may prefer the more precise description using coordinates given here:

Now, what about Kazhdan–Lusztig–Vogan polynomials? I don’t really understand them, but let me give it a try.

Actually, I’ll just discuss the Kazhdan–Lustzig polynomials for an easier bunch of groups, namely the groups $\mathrm{A}_n$, also known as $PSL(n+1)$. This will already be enough to make most of you run away screaming. But you shouldn’t: these groups describe the symmetries of $n$-dimensional projective geometry. For $n = 2$, this is the geometry that governs the use of perspective in paintings!

In $n$-dimensional projective geometry we have points, lines, planes, and so on up to the $n$th dimension, and all we can talk about is whether a point lies on a line, or a point lies on a plane, or a line lies on a plane, etcetera. There are some axioms, a bit like the axioms of Euclidean geometry, but simpler, because the concepts of distance and angle are not present! After all, if you draw a painting of a scene from different perspectives, distances and angles on your painting will change — but the fact that two lines intersect at a given point will not.

As I explained back in week186, a flag in $n$-dimensional projective geometry is a point, lying on a line, lying on a plane, lying on a… and so on, up to the $n$th dimension. The set of all such flags is called the flag variety of $\mathrm{A}_n$.

If you pick one flag and call it your favorite, you can classify other flags by how they’re related to your favorite flag. For example, maybe the point of the other flag lies on the line of your favorite flag. Or maybe the plane of the other flag contains the point of your favorite flag. Or maybe both these things are true. There are lots of possibilities.

In fact, there turn out to be exactly $(n+1)!$ possibilities. I won’t explain why, but there are lots of beautiful proofs of this beautiful fact.

So, if we classify flags according to this scheme, the flag variety gets partitioned into $(n+1)!$ subsets. These are called Schubert cells.

The closure of a Schubert cell is called a Schubert variety.

We can define projective geometry, and thus the flag variety and Schubert varieties, over any field. Let use the finite field with $q$ elements. Then, if we fix two Schubert varieties, we can count the number of points in the intersection of their closures. This number will be some polynomial in $q$. This is called an $R$–polynomial. Since it depends on a choice of two Schubert cells, there are $(n+1)! \times (n+1)!$ of these, naturally arranged in a square matrix.

The Kazhdan–Lusztig polynomials can be defined as certain linear combinations of $R$-polynomials. They don’t count the points in the intersection of two Schubert varieties… they do something related, but subtler: they convey information about their intersection cohomology. Also, by the Kazhdan–Lusztig conjecture, subsequently proved by Brylinski, Kashiwara, Beilinson and Bernstein, they encode facts about the representation theory of the complex form of $\mathrm{A}_n$. A bit more precisely, they say stuff about how many times any given Verma module contains any given irreducible representation as a subquotient.

This is a quite deep subject — for some reason that’s what I always say when something is over my head. The so-called Weil conjectures (now theorems) play a role here, since they relate the number of points in a smooth variety over a finite field to the cohomology of the corresponding smooth variety over the complex numbers. But Schubert varieties fail to be smooth, so we need a tricky form of cohomology — intersection cohomology — and a generalization of the Weil conjectures. For more, try these:

• Francesco Brenti, Kazhdan-Lusztig polynomials: history, problems, and combinatorial invariance.
• Frances Kirwan, An Introduction to Intersection Homology Theory, Pitman Research Notes in Mathematics 187, Longman Scientific & Technical, 1988. See especially chapter 8: the Kazhdan–Lusztig conjecture.
• David Kazhdan and George Lusztig, Representations of Coxeter groups and Hecke algebras, Inv. Math. 53 (1979), 165–184.
• David Kazhdan and George Lusztig, Schubert varieties and Poincaré duality, in Geometry of the Laplace Operator, Proc. Symp. Pure Math. 36, Amer. Math. Soc. (1980), 185–203.
• Sergei Gelfand and Robert MacPherson. Verma modules and Schubert cells: a dictionary, in Seminaire d’algebre Paul Dubriel et M. P. Malliavin, Lecture Notes in Mathematics no. 925 (1982), Springer Verlag, 1–50.

Whew! A lot of work, and we’re still fairly far from the actual calculation that was just done!

First, we need to go from $\mathrm{A}_n$ to $\mathrm{E}_8$. Everything I’ve discussed for $\mathrm{A}_n$ generalizes to other simple Lie groups; $\mathrm{E}_8$ just happens to be the trickiest example of these. Just as $\mathrm{A}_n$ is the symmetry group of $n$-dimensional projective geometry, $\mathrm{E}_8$ is the symmetry group of a kind of geometry first studied in depth by Hans Freudenthal. Instead of points, lines, planes and so on, this geometry has 8 kinds of figures following certain rules of their own! We could define a ‘flag’ in this context to consist of one of each of the 8 kinds of figures, all incident to each other. We can then use this to define a flag variety, and Schubert varieties, and $R$-polynomials and Kazhdan–Lusztig polynomials for $\mathrm{E}_8$.

Second, the Kazhdan–Lusztig–Vogan polynomials are more subtle than the Kazhdan–Lusztig polynomials. They are defined not in terms of intersection cohomology, but directly in terms of representation theory. I mentioned that Kazhdan–Lusztig polynomials give information about Verma modules. Verma modules are among the easiest to understand of the infinite-dimensional representations of Lie groups. But there are lots of other infinite-dimensional representations, and the Kazhdan–Lusztig–Vogan polynomials give similar information about these. The ones that were just computed give information about the infinite-dimensional unitary representations of the ‘split real form’ of $\mathrm{E}_8$. They say how many times any given ‘standard’ representation contains any irreducible representation as a subquotient.

Okay. I’ll quit here before I lapse into jargon that I don’t really understand. At best I’ve given you a rough flavor of the ideas leading to what was actually done. Maybe now you’re ready to look at this:

To go deeper still, read this:

By the way: I’ve improved the original version of this blog entry with the help of corrections in the comments below. So, with any luck, most of the corrections you see there concern mistakes that I’ve fixed by now!

Posted at March 19, 2007 12:39 AM UTC

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### Re: News about E8

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. SLn(R) has a torus, the diagonal matrices or their connected component, which is a product of R+’s, but SUn 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 KC = 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

Posted by: David Ben-Zvi on March 19, 2007 4:42 AM | Permalink | Reply to this

### Re: News about E8

The news says they “map” E8… does this mean that one can use the Kazhdan-Lusztig-Vogan polynomial to calculate the multiplicities of “simples” in any respesentation, not just “standard” ones? So can it be used to recover the entire category of representations of the split real form of E8? Also, could one “reconstruct” the split real form of E8 from its Kazhdan-Lusztig-Vogan polynomial?

Posted by: Daniel Moskovich on March 22, 2007 9:48 AM | Permalink | Reply to this

### Re: News about E8

These polynomials give the transition matrix between two bases of the K-group (Grothendieck group) of the category of representations, one given by simples and the other by standards. So if you have a representation whose K-class you know in terms of standards you can tell the multiplicities of simples, and you can tell characters. One can also use this to describe the action of intertwining operators (the Hecke algebra) on the K-group.

In fact these polynomials give more than just these multiplicities - that’s just their value at q=1 — they give a “graded” or “mixed” enhancement of this.

However this is quite far from telling you the categorical structure, it’s only K-theoretic information – for example you can’t tell about Ext groups or deformations from this information. And it certainly doesn’t recover the split real form in a natural way, though to do that you just need the category of finite dimensional representations.

Soergel has a beautiful conjecture that gives a lot of information about the categorical structure - in fact it’s a very strong form of the local Langlands conjecture for the field R. One of the main tools in the subject (that I presume they’re using) is Vogan’s K-theoretic enhancement of Langlands’ classification of simples, and Soergel says how to lift this to the categorical level (as a kind of Koszul duality!) In the case of complex groups Soergel proved his conjecture, but for real groups it’s still wide open.

Posted by: David Ben-Zvi on March 22, 2007 2:41 PM | Permalink | Reply to this

### Re: News about E8

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

Posted by: David Corfield on March 19, 2007 3:22 PM | Permalink | Reply to this

### Re: News about E8

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 :-)

Posted by: thomas1111 on March 19, 2007 3:41 PM | Permalink | Reply to this

### Re: News about E8

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

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.

Posted by: John Baez on March 19, 2007 6:12 PM | Permalink | Reply to this

### Re: News about E8

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.

Posted by: David Speyer on March 19, 2007 6:35 PM | Permalink | Reply to this

### Re: News about E8

David Speyer wrote:

John Baez wrote:

”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.

Oh, whoops! You’re right. I got my description of the Kazhdan–Lusztig polynomials from here:

but I misread Theorem 10 on page 11 to be about the Kazhdan–Lusztig polynomials instead of the $R$-polynomials. The idea of ‘working backwards from the Lefschetz trace formula’ to get the Kazhdan–Lusztig polynomials is implicit in Theorem 7 on the previous page.

I’ll fix my blog entry eventually, based on your correction.

Thanks a lot! I just happen to be trying to learn this general sort of stuff — stuff about flag varieties over $\mathbb{C}$, and over $F_q$, and how they’re related by the Weil conjectures and various spinoffs thereof — so your comment was incredibly helpful. I haven’t found what you said stated so clearly and tersely anywhere else: if I had, I wouldn’t have screwed up!

Posted by: John Baez on March 19, 2007 6:40 PM | Permalink | Reply to this

### Re: News about E8

I think one reason for the complexity of the calculation is it doesn’t involve usual Schubert varieties, as far as I understand (hope I’m not off base here) — the Kazhdan-Lusztig polynomials built from those are much better understood and calculated (I think Mark Goresky’s homepage has info on calculating them?) — rather we are looking at the same kind of problem, but for orbits (or their closures) of other groups on the flag variety. The usual KL story has to do with (infinite dimensional) representations of the complex group E8, while this has to do with its real forms. So even the enumeration of orbits is a lot trickier, not to mention calculating their intersection cohomology.

Posted by: David Ben-Zvi on March 20, 2007 6:41 AM | Permalink | Reply to this

### Re: News about E8

(Note adding in proof: I realized after writing this comment that I’m basically duplicating what David Ben-Zvi said, but I figure I might as well post, since the comment is already written)

While you’re mostly correct, there’s one important inaccuracy in the above.

The polynomials David Speyer describes are KL polynomials, which measure the singularities of B orbits on the flag manifold of E8. These have actually been known for a while, and don’t require an extremely Herculean calculation (though I don’t recommend trying it on your laptop).

What was actually computed are Kazhdan-Lusztig-Vogan polynomials (which David Vogan insists on calling “KL polynomials” confusing the hell out of those of us who aren’t paying careful attention), for the real form of E8.

So, for every complex Lie group (for example, E8), there are a bunch of different real forms. Each group has a unique one which is compact (for example, the compact real form of SL(n,C) is SU(n)) and another which is as far from compact as possible, called the split form (for SL(n,C) this is SL(n,R)). The split form is distinguished by the fact that its maximal torus is a product of copies of R-{0} (as opposed to the compact form, where it will be a product of circles).

If you like the presentation of simple Lie complex algebras using Xi, Yi, and Hi, the split form is what you get taking this with real coefficients.

So, let’s call G the split real Lie group associated to E8, and GC the complex group associated to E8.

Inside G, we have a maximal compact subgroup K (in fact, this is unique up to conjugacy). Let KC be the complex group that K is the compact form of.

As an example, look at the case of SLn instead of E8. Then we have SL(n,R) as the split real form, SL(n,C) its complexification, SO(n,R) as the maximal compact of the split form, and SO(n,C) as its complexification.

So, let F be the flag manifold of GC (that is, GC/B, where B is a maximal solvable subgroup). Now, the action of KC on F has finitely many orbits, and what KLV polynomials measure is how these orbits fit together, and especially what the singularities of orbit closures look like (the same way that KL polynomials do for Schubert varieties).

As for why anyone cares…well, if you think about Harish-Chandra modules of G, and where they go under the localization functor of Beilinson and Bernstein, you’ll recognize that understanding the geometry of these orbits is very important for understanding how these Harish-Chandra modules play together.

Posted by: Ben Webster on March 22, 2007 7:08 PM | Permalink | Reply to this

### Re: News about E8

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:

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.

Posted by: John Baez on March 19, 2007 8:32 PM | Permalink | Reply to this

### Re: News about E8

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.

Posted by: CB on March 19, 2007 10:01 PM | Permalink | Reply to this

### Re: News about E8

$\mathrm{E}_8$ is a Lie group, i.e. an infinite group with a topology that makes it a manifold. There’s no way to think of it as a group of permutations of a finite set. We can think of it as a group of permutations of an infinite set, but it doesn’t have a finite set of generators. So, you won’t find a file out there listing its generators as permutations.

However, there are various finite groups associated with $\mathrm{E}_8$.

For example, we can take the equations defining $\mathrm{E}_8$ and interpret them over a finite field instead over the real or complex numbers. Then we get a finite group depending on the size of the finite field (which is some prime power $q = p^k$). As I hinted, this plays a certain role in the Kazhdan–Lusztig–Vogan polynomials.

But, I don’t know any files listing the generators of these finite groups.

If you like permutation groups, you’ll be happiest thinking about the Weyl group of $\mathrm{E}_8$. This is the group of rotation and reflection symmetries of the 8-dimensional polytope associated to $\mathrm{E}_8$:

The Weyl group of $\mathrm{E}_8$ has 696,729,600 elements. There’s a beautiful description of it in terms of generators and relations, and you can think of its elements as permutations of the 240 vertices of this polytope.

The Weyl group of $\mathrm{E}_8$ is also related to the Kazhdan–Lusztig stuff, since each element corresponds to a Schubert cell in the flag manifold of $\mathrm{E}_8$. That’s why you saw the same large number in my post.

Posted by: John Baez on March 20, 2007 5:02 AM | Permalink | Reply to this

### Re: News about E8

“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.

Posted by: joe on March 19, 2007 11:43 PM | Permalink | Reply to this

### Re: News about E8

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.

Posted by: urs on March 20, 2007 10:49 AM | Permalink | Reply to this

### Re: News about E8

Oh, now I see that H. Nicolai is even quoted on that Atlas project webpage, where it says:

According to Hermann Nicolai, Director of the Albert Einstein Institute in Bonn, Germany (not affiliated with the project), “This is an impressive achievement. While mathematicians have known for a long time about the beauty and the uniqueness of $E_8$, we physicists have come to appreciate its exceptional role only more recently — yet, in our attempts to unify gravity with the other fundamental forces into a consistent theory of quantum gravity, we now encounter it at almost every corner! Thus, understanding the inner workings of E8 is not only a great advance for pure mathematics, but may also help physicists in their quest for a unified theory.”

Posted by: urs on March 20, 2007 10:54 AM | Permalink | Reply to this

### Re: News about E8

And by the way, the webpage here contains a misprint:

According to Hermann Nicolai, Director of the Albert Einstein Institute in Bonn

The AEI that H. Nicolai is the director of is not in Bonn but in (Potsdam-)Golm, a small place close to Berlin. Means: Nicolai is a theoretical physicist, not a mathematician proper.

Posted by: urs on March 20, 2007 11:01 AM | Permalink | Reply to this

### Re: News about E8

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

Posted by: John Baez on March 20, 2007 6:26 PM | Permalink | Reply to this

### Re: News about E8

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:

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

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.

Posted by: Mike Rios on March 20, 2007 9:49 PM | Permalink | Reply to this

### Re: News about E8

You can read a description of the 57-dimensional manifold on which $\mathrm{E}_8$ acts here, near the end. Briefly, it arises from a $\mathbb{Z}$-grading of the Lie algebra $e_8$:

$e_8 \cong e_8(-2) \oplus e_8(-1) \oplus e_8(0) \oplus e_8(1) \oplus e_8(2)$

with

$[e_8(i) , e_8(j)] \subseteq e_8(i+j)$

where the dimensions go as follows:

$248 = 1 + 56 + 134 + 56 + 1$

Since

$[e_8(0) , e_8(0)] \subseteq e_8(0),$

it follows that $e_8(0)$ is a 134-dimensional Lie algebra. Anyone who remembers that the dimension of $e_7$ is 133 will be unsurprised to hear that $e_8(0)$ is the Lie algebra direct sum $e_7 \oplus u(1)$. Since

$[e_8(0), e_8(2)] \subseteq e_8(2)$

it follows that $e_8(2)$ provides a 56-dimensional representation of $e_7$, which is indeed the famous Freudenthal triple system.

Since $e_8(-2) \oplus e_8(-1) \oplus e_8(0)$ is closed under the Lie bracket, it gives rise to a Lie subgroup

$H \subset \mathrm{E}_8$

and the quotient space

$\mathrm{E}_8/H$

is a 57-dimensional manifold on which $\mathrm{E}_8$ acts, since

$dim(\mathrm{E}_8/H) = dim(e_8(2) + e_8(3)) = 57$

So, indeed, $\mathrm{E}_8/H$ can be thought of as having tangent spaces that look like the Freudenthal triple system plus a 1-dimensional extra space.

I would like to categorify this, and find a collection of 57 algebraic varieties corresponding to the canonical basis of $e_8(2) \oplus e_8(3)$. This would let me write a paper entitled ‘57 Varieties’, and get sued for trademark infringement by Heinz.

Posted by: John Baez on March 21, 2007 2:16 AM | Permalink | Reply to this

### Re: News about E8

I gave an explicit realization of e8 corresponding to this grading in section 4 of math-ph/0301006. e8 can be described as vector fields in a 57-dimensional space preserving certain forms α and βijkl up to functions, i.e. they preserve the “lightcone” α = βijkl = 0. This is analogous to how the conformal group in n dimensions so(n,2) (or so(n+2) since I work over the complex numbers) can be realized as vector fields in n dimensions, corresponding to the grading so(n+2) = Cn + (so(n)+C) + Cn.

I was simply overwhelmed by this wealth, which of course grows even worse for Kac-Moody algebras like e10, e11 and e4711.

Posted by: Thomas Larsson on March 21, 2007 9:02 AM | Permalink | Reply to this

### Re: News about E8

Thomas Larsson wrote about the plethora of gradings on $e_8$, using words such as:

[…] disillusioned […] Even worse

In which sense would this wealth of gradings a bad thing? Isn’t it rather an interesting and useful fact?

Posted by: nonamegiven on March 21, 2007 11:53 AM | Permalink | Reply to this

### Re: News about E8

Sure, a classification of all possible gradings of all simple Lie algebras would be interesting, and perhaps such a classification already exists. A list up to depth 2 (3-gradings and 5-gradings) can be found in

S. Kaneyuki, Graded Lie algebras, related geometric structures, and pseudo-hermitian symmetric spaces, in Analysis and geometry on complex homogeneous domains,103, eds. J. Faraut, S. Kaneyuki, A. Koranyi, Q.-K. Lu, and G. Roos (Birkhäuser 2000)

My point was rather that given how easy it is to construct many different gradings, it is hard to attach special interest to any specific one, without external physical input.

Following Victor Kac, I was once very enthusiastic about an exceptional infinite-dimensional Lie superalgebra called E(3|8), mainly because there is a 1-1 correspondence between E(3|8) irreps and su(3)+su(2)+u(1) irreps. I thought this was a remarkable property, until I realized that every Lie algebra whose Dynkin diagram has four dots in a row, i.e. su(5), sp(8), so(9) and f4 (in two ways), has the same property.

Posted by: Thomas Larsson on March 21, 2007 1:58 PM | Permalink | Reply to this

### Re: News about E8

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
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:

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

Jeff

Posted by: John Baez on March 21, 2007 3:10 AM | Permalink | Reply to this
Weblog: The n-Category Café
Excerpt: A huge new calculation involving the exceptional Lie group E8.
Tracked: March 21, 2007 4:00 AM

### Re: News about E8

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

Posted by: John Baez on March 21, 2007 4:10 AM | Permalink | Reply to this

### Re: News about E8

That article was somewhat painful to read. I do love the caption from that picture, though. “Jeffrey D. Adams and a Lie group”

Posted by: John Armstrong on March 21, 2007 6:21 AM | Permalink | Reply to this

### Re: News about E8

Hello.
Thank you.
So I introduce your article in my blog.

Posted by: NoSyu on March 21, 2007 9:29 AM | Permalink | Reply to this
Read the post Split Real Forms
Weblog: Musings
Excerpt: Adding to the media frenzy.
Tracked: March 22, 2007 6:26 PM
Read the post Philosophia Naturalis #8
Excerpt: This post was delayed by a number of ridiculous technical mishaps, but issue number eight of Philosophia Naturalis - the physics blogosphere’s very own blog carnival - is finally here. There were a number of very interesting submissions spanning ...
Tracked: March 30, 2007 9:18 PM
Read the post E8 Quillen Superconnection
Weblog: The n-Category Café
Excerpt: On Quillen superconnections on E8-bundles and on the mathematical interpretation of the connection appearing in an article by A. Lisi.
Tracked: May 10, 2008 9:51 AM

### Observed in supercoooled crystal? Re: News about E8

‘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

Posted by: Jonathan Vos Post on January 8, 2010 12:39 AM | Permalink | Reply to this

### Re: Observed in supercoooled crystal? Re: News about E8

Cool! Thanks for the pointer, Jonathan.

The Science abstract says:

It is not often that an exact theory can describe a many-particle quantum-mechanical system, but one of the few exceptions is the behavior of a string of ferromagnets—an Ising chain—at magnetic field strengths that separate different types of ordering. Its excitations were predicted 20 years ago to be governed by the symmetry group E8, one of the most intriguing objects in mathematics. Now, Coldea et al. (p. 177) report direct experimental confirmation of this result in a quasi-one-dimensional Ising ferromagnet CoNb2O6, which they probed by neutron scattering. Two of the eight predicted excitations could be observed. Moreover, the ratio of the two lowest excitations is in quantitative agreement with the so-called “golden ratio” predicted by theory.

The golden ratio is deeply related to the E8 root lattice, as explained here. I wonder if that’s relevant here.

Does anyone know these theoretical papers by Zamolodchikov? Or know an electronic way to obtain them? It seems these were the ones that connected $E_8$ to the Ising model:

• A.B. Zamolodchikov, Proc. Taniguchi Symp. (Kyoto, 1988), in: Advanced studies in pure mathematics; and Rutherford Appleton Laboratory preprint RAL-89-001 (1989).
• A. B. Zamolodchikov, Integrals of motion and S-matrix of the (scaled) $T=T_c$ Ising-model with magnetic-field, Int. J. Mod. Phys. A 4 (1989), 4235.
Posted by: John Baez on January 8, 2010 5:52 AM | Permalink | Reply to this

### Re: News about E8

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?

Posted by: isomorphismes on April 13, 2015 6:41 PM | Permalink | Reply to this

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