Lie Theory Through Examples 3

October 21, 2008

Posted by John Baez

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We spent last week catching up with the notes. I decided to spend this week’s seminar explaining how the concept of weight lattice, so important in representations of simple Lie groups and Lie algebras, connects to what we’ve been doing so far. My approach follows that of Frank Adams:

J. Frank Adams, Lectures on Lie Groups, University of Chicago Press, Chicago, 2004.

This book puts the representation theory of Lie algebras in its proper place: subservient to the Lie groups! At least, that’s the right way to get started. Groups describe symmetries; a Lie algebra begins life as a calculational tool for understanding the corresponding Lie group. Only later, when you become more of an expert, should you dare treat Lie algebras as a subject in themselves.

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Lecture 3 (Oct. 21) - Representations of Lie groups. The weight lattice of a simply-connected compact simple Lie group.

I’m busy preparing a calculus midterm and two talks I’ll be giving later this week at the University of Illinois at Urbana–Champaign, so no pretty pictures this time. Eventually the weights of representations will give us beautiful pictures… but not today!

(In Illinois I’ll be visiting Eugene Lerman and also the topologist Matt Ando. I’ll speak about the number 8 and also higher gauge theory and the string group. Regular customers know both these talks, but I decided to tweak the second one a bit, to make it easier to understand — mainly by leaving stuff out. So, if it was incomprehensible last time you tried, check it out now.)

Posted at October 21, 2008 6:26 AM UTC

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Re: Lie Theory Through Examples 3

Hi John,

So, can I expect that the most dense lattice, in relation to the Zⁿ lattice density (p. 37), is E₈? I mean, there are no exceptional groups bigger than E₈.

In (p. 35), you said that D₈ allows an extra space, so that you can use a denser lattice. But, in (p. 37), you give the number for denser lattices in 6 and 7 than you implied 2 pages before! Why?

Also, I read your exposition about “24”, but it was not clear why that was the densest. You didn’t show any comparative table as in the exposition about the “8”.

Posted by: Daniel de França MTd2 on October 21, 2008 12:28 PM | Permalink | Reply to this

Re: Lie Theory Through Examples 3

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Daniel wrote:

So, can I expect that the most dense lattice, in relation to the $Z^n$ lattice density (p. 37), is $E_8$ ?

I don’t think so.

You’re right about one thing: the fun way to compare densities of sphere packings between different dimensions is not to say what percent of space is filled by spheres — as I did in my talk. Instead, it’s best to build the packing with spheres of radius one and then count how many centers of spheres there are per unit volume. This ratio is called the ‘center density’ by Conway and Sloane.

For example, if we pack unit-radius spheres as densely as possible in 1 dimension, the center density is $1/2$ . Warning: not 1, but $1/2$ !

Similarly, if we pack unit radius spheres in a hypercubical lattice in $n$ dimensions, the center density is $1/2^n$ . That’s because this lattice is not $\mathbb{Z}^n$ , but $(2 \mathbb{Z})^n$ .

You can find lots of information about center densities of lattices in Conway and Sloane’s magnificent book, Sphere Packings, Lattices and Groups. Every mathematician should look at this amazing book!

You can also find a lot of information about center densities in Sloane’s online Catalogue of Lattices.

For example, the center density for $E_8$ is $1/16$ . That’s a lot worse than the center density of the Leech lattice in 24 dimensions! The Leech lattice has center density 1 — the best possible center density of any lattice in dimension $\lt 30$ .

But, in (p. 37), you give the number for denser lattices in 6 and 7 than you implied 2 pages before! Why?

Those are the $E_6$ and $E_7$ lattices, which arise most naturally as certain ‘slices’ of the $E_8$ lattice. Read week65 (or Conway and Sloane’s book) for details.

Also, I read your exposition about “24”, but it was not clear why that was the densest. You didn’t show any comparative table as in the exposition about the “8”.

I got lazy. If you make a nice table in TeX, I’ll stick it in the paper before I publish it, and credit you. All the necessary information is in Sloane’s online catalogue, or the book by Conway and Sloane.

Posted by: John Baez on October 21, 2008 9:36 PM | Permalink | Reply to this

Re: Lie Theory Through Examples 3

“Those are the E 6 and E 7 lattices, which arise most naturally as certain ‘slices’ of the E 8 lattice.”

Sure, that was on the footnotes. But I what meant is that on p.34, it seems that your justification to stop using Dn was that since one more sphere could be packed once arriving in 8 dimensions, you should use E8. So, I was wondering why you said that given that in the table there were lattices with higher densities, related to E8, not to Dn, as I’d expect.

“Similarly, if we pack unit radius spheres in a hypercubical lattice in n dimensions, the center density…”

I noticed the pattern of the inverse of power of 2, and because of that, I was surprised that the biggest number appeared divided the E8 density by the Zn density in the table. I thought that given the fame nobility of E8, there could be also something special about the density of E8. But I think it’s a coincidence, is it?

Posted by: Daniel de França MTd2 on October 22, 2008 4:07 AM | Permalink | Reply to this

Re: Lie Theory Through Examples 3

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Daniel wrote:

But I what meant is that on p.34, it seems that your justification to stop using $D_n$ was that since one more sphere could be packed once arriving in 8 dimensions, you should use $E_8$ . So, I was wondering why you said that given that in the table there were lattices with higher densities, related to $E_8$ , not to $D_n$ , as I’d expect.

It just happens to be incredibly easy to describe the $E_8$ lattice starting from the $D_8$ lattice: it consists of two copies of the $D_8$ lattice, one shifted with respect to the other. The $E_6$ and $E_7$ lattices are not constructed this way — they’re not so quite so easy to describe.

In particular, the $E_8$ lattice is twice as dense as the $D_8$ lattice. But the $E_7$ lattice is not twice as dense as the $D_7$ lattice, and similarly for $E_6$ and $D_6$ .

Remember, this was a talk — a talk that was supposed to be lots of fun. So, I only wanted to talk about stuff that was very simple and beautiful. I wasn’t aiming for thoroughness!

I thought that given the fame and nobility of $E_8$ , there could be also something special about the density of $E_8$ . But I think it’s a coincidence, is it?

The $E_8$ lattice has lots of nice features, but as you can see from Sloane’s chart, its center density is not amazingly high. Look at the center density of the densest known lattice in 128 dimensions!

Posted by: John Baez on October 22, 2008 6:25 AM | Permalink | Reply to this

Re: Lie Theory Through Examples 3

The E 8 lattice has lots of nice features, but as you can see from Sloane’s chart, its center density is not amazingly high. Look at the center density of the densest known lattice in 128 dimensions!

That’s true, but center density is actually a problematic way to compare lattices in different dimensions. It’s normalized in such a way that it automatically grows incredibly fast (like a factorial) even for mediocre packings in high dimensions, and it is difficult to make a meaningful comparison between dimensions.

The main advantage of center density compared to the packing density is just that the numbers look simpler, since factors of pi and big factorials have been removed.

Posted by: Henry Cohn on October 24, 2008 4:36 AM | Permalink | Reply to this

Re: Lie Theory Through Examples 3

” a problematic way to compare lattices in different dimensions.”

So, is it possible to define a lattice density that actualy by itself make E8 look special and cool, in a meaningful way?

Posted by: Daniel de França MTd2 on October 24, 2008 11:47 AM | Permalink | Reply to this

Re: Lie Theory Through Examples 3

It just happens to be incredibly easy to describe the E_8 lattice starting from the D_8 lattice: it consists of two copies of the D_8 lattice, one shifted with respect to the other.

Put differently, one can locate the D_8 lattice inside the E_8, and it’s index two. (Of course that’s not the viewpoint that you take when constructing it.)

I think this just amounts to taking the Dynkin diagram, affinizing, and erasing a vertex, which finds a sub-root-system and hence a sublattice. The corresponding subgroup is the centralizer of an element of (adjoint) order = the coefficient of the corresponding simple root in the high root = the index of this sublattice. (This is this Borel-de Siebenthal theory that seems to be the main thing I ever post about here.)

In the E_8 case, we affinize, then erase the vertex two steps away from the trivalent vertex. It is labeled by 2. The result is the D_8 diagram.

The E_6 and E_7 lattices are not constructed this way — they’re not so quite so easy to describe.

From the point of view espoused above, the E_6 is made out of two copies of the A_1 x A_5 lattice, or three of the (A_2)^3 lattice. Whereas the E_7 is made out of two copies of the A_7 lattice, or two copies of the A_1 x D_6, or three of the A_2 x A_4, or four of the A_3 x A_1 x A_3.

Posted by: Allen Knutson on November 25, 2008 10:48 PM | Permalink | Reply to this

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