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

In the long process that resulted in the classification of finite simple groups, some of the exceptional groups were only shown to exist after people had computed (most of) their c …

Among simple Lie groups, $Spin(8)$ is the most symmetrical one in the sense that $Out(Spin(8))$ is the largest possible group. A description of this outer automorphism groups is as …

It is known that for a simple algebraic group over an algebraically closed field of positive characteristic (which I assume to be {\it good} for the group), the Weyl modules corres …

I am interested in conjugacy classes in connected reductive groups over a non-archimedean field $F$ of characteristic $0$, or its algebraic closure. On this topic it is often requi …

In a recent question Deane Yang mentioned the beautiful Riemannian geometry that comes up when looking at $G_2$. I am wondering if people could expand on the geometry related to th …

To study the topology of Lie groups, you can decompose them into the simple compact ones, plus some additional steps, such as taking the cover if necessary. After that, the structu …

Recall the list of irreducible simply connected inner symmetric spaces of compact type in dimension $4k+2$: Hermitian symmetric spaces (one can write them down explicitly); Grass …

There are three exceptional Galois groups $L_2(5)$, $L_2(7)$ and $L_2(11)$ . These are cited as one of Arnold's "trinities" and are connected with other trinities and the McKay Cor …

For this question I'm happy to take the complex numbers as the base field. I've been trying to learn a little bit about the exceptional Lie algebras and for a while they seemed in …