18
votes
Groups that do not exist
I believe that at some point there was a conjecture (by whom, I don't recall) that Janko's smallest group, of order $175,560=11(11^2-1)(11^3-1)/(11-1)$, should be the first of an infinite sequence of ...
- 2,709
17
votes
Does the 3875-dimensional rep of $E_8$ have a solution to $x\star x=0$?
Yes.
The basic representation of $E_8$ has character $j(\tau)^{1/3} = q^{-1/3}(1+248q+4124q^2 + \cdots)$, and the 4124 decomposes as $1+248+3875$. By Frenkel-Kac-Segal, the basic representation has ...
- 45.7k
15
votes
Accepted
Where can I find details of Elie Cartan's thesis?
$\mathrm{G}_2$ is the only one of the exceptional groups that can be defined as the stabilizer of a `generic' tensorial object on a vector space and, over the complex numbers, even this is not quite ...
- 108k
13
votes
Accepted
Constructing $E_8$ from its branching to $A_8$
An excellent reference for this kind of descriptions (and for many other facts about $E_8$!) is Skip Garibaldi's paper "$E_8$, the most exceptional algebraic group" in the Bulletin of the AMS (here).
...
- 6,614
12
votes
Accepted
Does $G_2(\mathbb{Z})$ depend on the choice of an integral model?
The short answer is no, because a Chevalley group has infinitely many $\mathbb{Z}$-points (even Zariski dense by the Borel density theorem).
For the long answer, let me first completely describe all $\...
- 984
12
votes
Accepted
What is the largest subgroup of $GL^{+}(7,\mathbb{R})$ which smoothly retracts onto $G_2$?
There is no retraction of $\mathrm{SO}(7)$ onto $\mathrm{G}_2$. If such a retraction $\rho:\mathrm{SO}(7) \to \mathrm{G}_2$ existed, then the composition
$$
\mathrm{G}_2 \hookrightarrow \mathrm{SO}(7)...
- 108k
10
votes
Constructing $E_8$ from its branching to $A_8$
Of course, the original description of this branching is due to Élie Cartan, himself. For example, see Chapitre IX of his 1914 paper Les groupes réels simples, finis et continu (Annales scientifiques ...
- 108k
9
votes
Relation between different $E_8$ matrices
$M_1$, $M_4$ and $M_5$ are all the same quadratic form up to integer change of basis: To get from $M_1$ to $M_4$, conjugate by the diagonal matrix with digaonal entries $(1,-1,1,-1,1,-1,1,-1)$; to get ...
- 156k
9
votes
Accepted
Do the exceptional root systems arise in the real world?
As Noam Elkies alluded to in a comment, the E8 lattice plays a role in coding theory, basically because it such an efficient sphere packing. For example, Kurkoski has proposed using it for error ...
- 82.6k
8
votes
Concrete description of an exceptional minuscule variety
One description is $\smash{X=E_7\left/E_6\times S^1\right.}$ (quotient of compact groups, of real dimension 133 – 79 = 54), as removal of the root $\alpha$ in question leaves an $E_6$ diagram. Another ...
- 31.5k
8
votes
Flag manifolds as incidence correspondences
The "incidence" relation (at least for types $E_6$ and $E_7$) for all pairs of vertices $i$, $j$ is described in S. Garibaldi, M. Carr "Geometries, the principle of duality, and algebraic groups", ...
- 1,602
7
votes
Accepted
Quadratic forms on $\mathbb{R}^{16}$ coming from octonions
I'm revising my answer because whether the image of the map $j$ that the OP defines is equal to the $10$-dimensional subspace $H$ of $\mathrm{Sym}^2(\mathbb{R}^{16})$ that is invariant under $\mathrm{...
- 108k
7
votes
Accepted
Viewing exceptional Lie algebras via the classical ones
Élie Cartan himself, recognized and used the following description of $\mathfrak{e}_6$: Let $V$ be a vector space of dimension $6$ and let $W$ be a vector space of dimension $2$. Then there is a ...
- 108k
7
votes
Accepted
How to describe the compact real forms of the exceptional Lie groups as matrix groups?
Cartan describes all of the compact real forms of the simple Lie groups over $\mathbb{C}$ in his first paper that classifies the real forms. In fact, he describes them exactly in the terms that you ...
- 108k
6
votes
Accepted
A question on complex semisimple Lie groups and $(\mathbb{C}^2, \omega)$
I think that the kind of question you are asking is one that was treated by Dynkin back in the 1950s (see Semisimple subalgebras of semisimple Lie algebras. (Russian) Mat. Sbornik N.S. 30(72), (1952), ...
- 108k
6
votes
Accepted
Why do these two irreps of $E_6$ have the same dimension?
$\newcommand\Sym{\mathrm{Sym}}$
An extended comment which more or less suggests that your suggested answer might be as good as one can do.
If $G$ has a representation on $V$ which preserves a ...
- 76
6
votes
Accepted
Orbits of action of the split group of type $F_4$
I work over the complexe numbers, but the story is similar over any fields (provided you choose the split octonions).
Let us identify the Albert algebra $A$ with the algebra of self-adjoint $3*3$ ...
- 7,300
5
votes
Description of the generalized grassmannians and flag varieties (parabolic quotients) associated to the exceptional groups
Edit 2: A good discussion of the lowest dimensional pieces of each of the flags below is found in Geometries, the principle of duality, and algebraic groups by Carr and Garibaldi. In particular for ...
- 954
5
votes
Diagonalization of octonionic Hermitian matrices of size $2\times 2$
Yes, in fact, any $2$-by-$2$ octonionic Hermitian matrix is equivalent under the natural $\mathrm{Spin}(9)$ action to a diagonal $2$-by-$2$ octonionic Hermitian matrix.
This follows from the well-...
- 108k
5
votes
Relation between different $E_8$ matrices
A euclidean lattice can be defined by a generating basis, for example as a nonsingular $n \times n$ real matrix $F$ whose columns generate the lattice which consists of all vectors $Fx$, $x \in \...
- 1,362
4
votes
How to describe the compact real forms of the exceptional Lie groups as matrix groups?
There is an abstract way of integrating Lie algebras but I guess you are asking for a more hands on approach. I suggest browsing Exceptional Lie groups by Ichiro Yokota. Usually, it's the compact (or ...
- 8,597
4
votes
How to check whether a given matrix is in the image of a representation?
Suppose that $G$ is a compact Lie group (or, more generally, an algebraic group over some field $F$), ${\mathfrak g}$ is its Lie algebra. You are given a linear representation $\rho: G\to GL(n, F)$. ...
- 12.2k
3
votes
Accepted
About certain elements in the zero weight space of an irreducible representation of the complex simple Lie algebra of type G$_2$
The answer is no. Here is some SAGE code:
...
- 8,866
3
votes
Why do these two irreps of $E_6$ have the same dimension?
A geometric picture of the answer given by user484566 is that (certain real form of) $E_6$ is octonionic version of the special linear group $SL_3$. The symmetric three-tensor is the polarization of ...
- 8,597
3
votes
Where can I find details of Elie Cartan's thesis?
All details about Cartan's thesis can be found in the thesis itself:
https://archive.org/details/surlastructured00bourgoog
There is also a German translation for those who do not read French:
Ueber ...
- 91.8k
3
votes
Constructing real forms of the Tits-Freudenthal magic square for (Rosenfeld) projective planes
Alberto Elduque obtained all real forms of exceptional Lie algebras in his variant of Freudenthal square in A new look at Freudenthal's magic square, Non-associative algebra and its applications, 149–...
- 8,597
3
votes
Embed exceptional non-compact simply connected simple Lie groups into classical simple Lie groups with preserving centers
This fails already for the split form of $\mathrm{G}_2$. Every finite-dimensional irreducible representation of its Lie algebra is a constituent of a tensor power of the $7$-dimensional representation....
- 108k
1
vote
How to check whether a given matrix is in the image of a representation?
I can answer the diagonal case, at least, where the answer is fairly easy. This assumes that you have everything set up 'nicely', which I'll define. I'm going to do the case of an algebraic group, as ...
- 1,143
1
vote
Accepted
$8 \times 31 = 8 \times 31$?
The decompositions are not conjugate. Pick (any) two Cartans in the direct sum decomposition. For the decomposition coming from $2^8 \subset E_8$, two Cartans together generate a subalgebra isomorphic ...
- 54.6k
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