Welcome to MathOverflow

Question

4

2

What is the smallest simplest(non-trivial) $E_8$ -module ?

edited Apr 9 at 18:51

asked Jan 19 at 12:40

Hassan Jolany
1,046●1●2●12

1

$\mathbb C\ \ \ $ – Allen Knutson Jan 19 at 15:20

6

I agree with Hassan; the smallest simplest $E_8$-module is actually $(0)$. – Robert Bryant Jan 19 at 16:11

1

Hassan's question combined with his comment makes the whole thing look trivial indeed. Anyway, the smallest faithful representations of all simple Lie algebras are worked out in the old literature and not very hard to relate to the fundamental weights. What is the point of the question? (And why the extra tags?) – Jim Humphreys Jan 19 at 16:22

2

This is somewhat beside the point, but the difference between Allen Knutson's and Robert Bryant's answers seems to hinge on whether the word "simple" is meant in a strict mathematical sense. Since $(0)$ is a unit under direct sum, it is often more convenient to adopt the convention that it is a non-simple module. – S. Carnahan♦ Jan 20 at 0:21

1

Zuckerman calls these "the trivial representation" and "the pre-trivial representation". – Allen Knutson Jan 20 at 21:43

show 2 more comments

Dietrich Burde · Accepted Answer · 2013-04-08 20:44:39Z UTC

Let $\mu(L)$ denote the minimal dimension of a faithful module of $L$. The complex simple Lie algebra $E_8$ satisfies $\mu(L)=\dim (L)$, as Cartan showed. Indeed, $E_8$ is the only complex simple Lie algebra with this property. There are more results in this direction (which are perhaps interesting): suppose that $L$ is a complex semisimple Lie algebra satisfying $\mu(L)=\dim (L)$. Then $L\simeq E_8\oplus \cdots \oplus E_8$. Even more general, let $L$ be a Lie algebra, where the solvable radical $rad(L)$ is abelian. Then always $\mu(L)\le \dim (L)$, and equality holds if and only if $L$ is abelian of dimension less than $5$, or $L$ is isomorphic to $E_8\oplus \cdots \oplus E_8$. (For references see arXiv:1006.2062).

Robert Bryant · Answer 2 · 2013-01-19 13:46:47Z UTC

Cartan showed that the lowest dimensional (nontrivial) $E_8$-module is ${\frak{e}}_8$ itself, i.e., the adjoint representation, which has dimension $248$. The next smallest nontrivial irreducible module is considerably larger dimension, $3875$, and I think that the next one after that has dimension $30380$.

At http://www-math.univ-poitiers.fr/~maavl/LiE/form.html, you can check these online using LiE.

smallest simplest $E_8$ -module

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

2 Answers

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.