Kostant on E_{8}
Posted by John Baez
At Riverside we recently heard a fascinating talk:

Bertram Kostant, On Some Mathematics in Garrett Lisi’s ‘E8 Theory of Everything’, U. C. Riverside, February 12th.
Abstract: A physicist, Garrett Lisi, has published a highly controversial, but fascinating, paper purporting to go beyond the Standard Model in that it unifies all 4 forces of nature by using as gauge group the exceptional Lie group $E_8$. My talk, strictly mathematical, will be about an elaboration of the mathematics of $E_8$ which Lisi relies on to construct his theory.
Luckily we had a video camera on hand. So, at the above link you can see streaming or downloadable videos of Kostant’s talk, as well as lecture notes.
Kostant’s talk was quite technical! If it’s too tough, you might prefer the warmup talk I gave earlier that day. But, Kostant described some ideas whose charm is easy to appreciate:
 The dimension of $E_8$ is $248 = 8 \times 31$. There is, in fact, a natural way to chop up $E_8$ into 31 spaces of dimension 8.
 There is a nice way to see the product of two copies of the Standard Model gauge group sitting inside $E_8$.
 The Standard Model gauge group is a subgroup of $SU(5)$. There is also a nice way to see the product of two copies of $SU(5)$ sitting inside $E_8$.
 The dimension of $SU(5) \times SU(5)$ is 48, and $248 48 = 200$. The adjoint action of $SU(5) \times SU(5)$ on the Lie algebra of $E_8$ thus gives a $200$dimensional representation, and this is $(5 \otimes 10) \oplus (\overline{5} \otimes \overline{10}) \oplus (10 \otimes 5) \oplus (\overline{10} \otimes \overline{5})$ [Actually this is wrong; see the discussion below.]
Garrett Lisi’s ideas have received serious criticism from Jacques Distler and others. I’ve included links to Lisi’s paper and also Distler’s comments. But, the work Kostant presents here is logically independent — beautiful math, regardless of its possible applications to physics. It makes heavy use of recent work on certain finite subgroups of $E_8$, most notably $GL(2,F_{32})$ and $(\mathbb{Z}/5)^3$. As Kostant said, “$E_8$ is a symphony of twos, threes and fives”.
Re: Kostant on E8
Grr! It seems like the downloadable video website is down again! It was working last night…