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-1
votes
1answer
193 views

Decomposition of $S^7=Spin(7)/G_2$

The seven sphere can be written as the reductive space $S^7=Spin(7)/G_2$. Has the decomposition $Spin(7)=G_2\times S^7$ been calculated somewhere; maybe in terms of Cayley numbers?
5
votes
1answer
236 views

Construction of Thom-Spectrum for G_2-Structures

The motivation to this question is the paper of Crowley and Nordstrøm "A New Invariant of $G_2$-Structures". I am trying to find a homotopy theoretic interpretation of the following geometric ...
2
votes
2answers
420 views

smallest simplest $E_8$ -module

What is the smallest simplest(non-trivial) $E_8$ -module ?
13
votes
4answers
1k views

Triality of Spin(8)

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 follows. $Spin(8)$ ...
43
votes
3answers
2k views

Groups that do not exist

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 character tables and ...
3
votes
3answers
394 views

Irreducibility of fundamental Weyl modules

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 corresponding to the ...
1
vote
2answers
267 views

Is the derived group of $G_2$ simply connected?

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 required that the group ...
5
votes
1answer
512 views

Is there a connection between exceptional Galois groups and Ramanujan's partition congruences

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 Correspondence. ...
27
votes
5answers
2k views

$G_2$ and Geometry

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 the exceptional Lie ...
20
votes
2answers
1k views

Does $\mathrm{E}_7/(\mathrm{SU}_8/(\mathbb{Z}/2))$ carry an almost complex structure?

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); Grassmannians of oriented ...
6
votes
2answers
449 views

How do Jordan algebras help one understand representations of exceptional Lie algebras?

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 inaccessible. I looked ...
19
votes
6answers
929 views

Explanation for E_8's torsion

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 structure of $SO(n)$'s is ...