# Questions tagged [exceptional-groups]

Exceptional Lie groups G2, F4, E6, E7, E8 of dimensions 14, 52, 78, 133, 248 were obtained as result of classification of simple Lie groups performed by Killing and Elie Cartan. The tool used in classification is Dynkin diagram and root system of vectors in Lie algebra of the group. The remaining Lie groups form four infinite families of transformations of n-dimensional space over real (odd and even), complex and quaternionic field.

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### How to describe the compact real forms of the exceptional Lie groups as matrix groups?

I know that $G_2$ can be described as the subgroup of $SO(7)$ preserving a specific element of $\Lambda^3(\mathbb{R}^7)^*$. It can thus be realized as a matrix group. Prof. Robert Bryant did describe ...
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### How to check whether a given matrix is in the image of a representation?

Let $G$ be a compact simple Lie group, and let $\rho$ be a (faithful, unitary) irreducible representation thereof of $\mathbb K$-dimension $n$, where $\mathbb K=\mathbb C/\mathbb R/\mathbb H$ if $R$ ...
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### Reference request: Commutator relations for the exceptional group F4

Is there any standard reference for the commutator relations for the exceptional group of type $F_4$? If this question is not appropriate here, please let me know and I will delete it. Thanks in ...
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### Orbits of action of the split group of type $F_4$

Let the split group of type $F_4$ act as the automorphism group of the split Albert algebra $A$. Consider the action of $F_4\times \mathbb{G}_m$ on $A$, given by letting $\mathbb{G}_m$ act by scalar ...
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### Better names for Lie groups

After reading this question I was wondering whether mathematicians tried to invent better names for exceptional simple Lie groups $F_4, E_6, E_7, E_8$ ? These names seems a bit obscure and does not ...
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### 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?
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### 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 ...
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### smallest simplest $E_8$ -module

What is the smallest simplest(non-trivial) $E_8$ -module ?
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### 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. $Spin(8)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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. ...
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### $G_2$ 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 ...
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### 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 ...
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### 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 ...
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### Geometric interpretation of exceptional Symmetric spaces

Elie Cartan has classified all compact symmetric spaces admitting a compact simple Lie group as their group of motion.There are 7 infinite series and 12 exceptional cases. The exceptional cases are ...
### Matrix representation for $F_4$
Has anyone ever bothered to write down the 26-dimensional fundamental representation of $F_4$? I wouldn't mind looking at it. Is it in $\mathfrak{so}(26)$? I'm familiar with the construction of the ...
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 ...