The exceptional-groups tag has no usage guidance.

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### A representation of Spin(9,1)

Let $Spin(9,1)$ denote the universal (double) cover of $SO(9,1)$. $Spin(9,1)$ acts linearly on $\mathbb{R}^{16}$ (see e.g. p.29 here https://arxiv.org/pdf/math/0105155v4.pdf ).
Consider the induced ...

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### Decomposition into irreducible components of a representation of $Spin(9)$

It is well known that the group $Spin(9)$ acts linearly on the vector space $\mathbb{R}^{16}$ (see for example "Spinors and calibrations" by R. Harvey).
Consider the induced representation of $Spin(9)...

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### Explicit generators of the Lie algebra $spin(9)$

It is well known that the Lie group $Spin(9)$ acts on the vector space $\mathbb{R}^{16}$ (see e.g. Harvey's book "Spinors and calibrations".) It is convenient to identify this vector space with the ...

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### Transitivity of $Spin(7)$ in triples of vectors

I have a simple question: transitivity of $Spin(7)$ in triples of orthogonal vectors. Let $Spin(7)\subset SO(8)$ act on $\mathbb{R}^8$, and $e_1,e_2,e_3$, $v_1,v_2,v_3$ be two triples of mutually ...

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### The special embedding $\mathfrak{so}(7)\subset\mathfrak{so}(8)$

It is commonly known that we have a chain of embeddings
$$SU(4)\subset Spin(7)\subset SO(8)$$
(there is more than one possible $Spin(7)$, just take one).
Which is the explicit analog for the Lie ...

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### Is this characterization of (-1)-eigenspaces of the Weyl group of $E_6$ known?

I recently needed to know which circles $S$ in a maximal torus $T^6$ of the compact exceptional group $E_6$ yield one-dimensional subspaces $\mathfrak s$ of the Lie algebra $\mathfrak t^6$ that are ...

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### How the exceptional simple Lie groups/ algebras were first discovered and by whom?

I am wondering whether exceptional simple Lie groups/ algebras were first discovered in order to obtain a complete list of such objects, or they appeared as answers to completely different questions.
...

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### $Spin(7)$ as stabilizer of a $4$-form revisited

For a better understanding of this question, please see the question and answer here.
In $Spin(8)$ there are plenty of copies of $Spin(7)$; consider, for instance, the antiimage of $SO(7)<SO(8)$ ...

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### $Spin(7)$ as stabilizer of a $4$-form

According to Bryant's work on special holonomy groups, $G_2\subset SO(7)$ may be defined as the group preserving the following 3-form:
$\phi_0=\mathrm{d}x_{123}+\mathrm{d}x_{145}+\mathrm{d}x_{167}+\...

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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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### 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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### 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 ...

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### 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 ...