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.
65
questions
6
votes
1
answer
360
views
Do the exceptional root systems arise in the real world?
I am looking for a list of real world examples where the exceptional roots systems $E_6, E_7, E_8, F_4$, and $G_2$, and their associated Lie algebras and Lie groups, arise. To make this question a ...
7
votes
1
answer
328
views
Relation between different $E_8$ matrices
There are several rank-8 square matrices known to be related to $E_8$:
Cartan $E_8$ matrix https://en.wikipedia.org/wiki/E8_(mathematics)#Cartan_matrix
$$M_1=\left [\begin{array}{rr}
2 & -1 &...
- 317
1
vote
1
answer
121
views
About certain elements in the zero weight space of an irreducible representation of the complex simple Lie algebra of type G$_2$
$\newcommand{\fg}{\mathfrak g}\newcommand{\ee}{\varepsilon}$Let $\fg$ be the complex simple Lie algebra of type G$_2$.
We consider its root system as follows (though it is probably not necessary to ...
- 2,269
0
votes
0
answers
82
views
Is there a formula for $fgf^{-1}$ in $G_2$?
Let $G_2$ be the exceptional Lie group of smallest dimension, represented as the exponentials of the antisymmetric real $7×7$ matrices $A$ with $$\sum_{i=1}^{3}{A(x+2^i,x-2^{i-1})} = 0$$ for integer $...
- 1,121
2
votes
0
answers
106
views
Do Weyl groups generate the exceptional Lie groups as sequences of reflexions in the Weyl chambers?
Platonic groups of symmetry are Weyl groups for the exceptional Lie algebra E6->E8, as root systems. These can be viewed as mirrors in a kaleidoscope (Goodman).
I would like to know if one can ...
9
votes
2
answers
341
views
Why do these two irreps of $E_6$ have the same dimension?
$E_6$'s Dynkin diagram is a line of 5 vertices, which we will number 1...5, and a sixth one attached to #3, which we will ignore.
$\dim V_{\omega_2} = 351 = \dim V_{2\ \omega_1}$, where $\omega_i$ ...
- 27.4k
7
votes
2
answers
415
views
Quadratic forms on $\mathbb{R}^{16}$ coming from octonions
$\DeclareMathOperator\RRe{Re}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sym{Sym}$Let $\mathcal{H}_2(\mathbb{O})$ denote the (10-dimensional) real vector space of octonionic Hermitian matrices ...
- 20.5k
14
votes
1
answer
278
views
Description of the generalized grassmannians and flag varieties (parabolic quotients) associated to the exceptional groups
If $G$ is a classical semisimple algebraic/Lie group over an algebraically closed field (maybe just say $\mathbb{C}$), viꝫ. $\mathit{SL}_n$, $\mathit{SO}_n$, $\mathit{Sp}_n$ (isogenies irrelevant here)...
- 28.2k
4
votes
0
answers
214
views
What is known about the cohomology of the U-duality group?
$\newcommand{\Es}{E_{7(7)}}\newcommand{\Z}{\mathbb Z}$Let $\Es$ denote the split form of $E_7$, which is a real Lie
group. It can be characterized as the subgroup of $\mathrm{Sp}_{56}(\mathbb R)$ ...
- 6,716
2
votes
1
answer
113
views
Diagonalization of octonionic Hermitian matrices of size $2\times 2$
The group $Spin(9)$ is a subgroup of $SO(16)$ and acts transitively on the unit sphere $S^{15}$. $Spin(9)$ acts naturally on the space of octonionic Hermitian $2\times 2$-matrices (I do not define ...
- 20.5k
10
votes
1
answer
385
views
Viewing exceptional Lie algebras via the classical ones
I've been trying to understand the exceptional Lie algebras through the classical ones that I am more familiar with. In particular I wanted to get a handle on the root spaces and most discussions that ...
- 867
4
votes
0
answers
109
views
Is there a smooth Weyl equivariant map from this quotient space into $G_2/T^2$?
It is known that $G_2$ acts transitively on $S^6$ with fibers $SU(3)$. Let us consider the following set $P$ of complex unitary $7 \times7$ matrices $A$, where
$$ A = (v_0, \, v_1, \, v_2, \, v_3, \, ...
- 4,697
2
votes
0
answers
67
views
Lie Groups and Lie algebras related to Jordan algebras
Let $J$ be a Jordan algebra. I knew three relative Lie groups/Lie algebras to $J$.
In the paper "The Capelli Identity, Tube Domains, and the Generalized Laplace Transform"
Jacobson [J] has ...
- 719
8
votes
1
answer
326
views
Does $G_2(\mathbb{Z})$ depend on the choice of an integral model?
I am trying to understand constructions of exceptional groups of type $G_2$ (over rings). In this post, by a model (of type $G_2$) I mean an affine smooth group scheme over $\mathbb{Z}$ such that the ...
- 1,504
4
votes
0
answers
72
views
Parabolic Bruhat graphs for exceptional types
I am looking for some computer software or a reference for some parabolic Bruhat graphs. In particular, what I really need $E_8 \setminus E_7$. Does anyone know where or how I'd find this?
- 1,191
4
votes
2
answers
467
views
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 ...
- 4,697
6
votes
2
answers
261
views
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 $R$ ...
1
vote
0
answers
100
views
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 ...
- 940
16
votes
0
answers
467
views
$8 \times 31 = 8 \times 31$?
The Lie algebra $\mathfrak{e}_8$ has (at least) two ways to be written as a direct sum of $31$ Cartan subalgebras.
First, Thompson and Smith showed that the (compact or complex) Lie group $\mathrm{E}...
- 52.2k
6
votes
0
answers
245
views
Branching rules for E6 into SU(3)^3
I am very confused about what are the branching rules for representations of $E6$ into a $SU(3)\times SU(3)\times SU(3)$ subgroup. At least in the physics literature, there seems to be a serious ...
- 469
22
votes
2
answers
597
views
Does the 3875-dimensional rep of $E_8$ have a solution to $x\star x=0$?
Consider the compact Lie group $E_8$. Its second-smallest fundamental representation is $3875$-dimensional and admits a symmetric invariant form, and so is real: $E_8 \curvearrowright \mathbb{R}^{3875}...
- 52.2k
8
votes
1
answer
219
views
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 ...
- 113
5
votes
0
answers
385
views
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 ...
user21230
12
votes
1
answer
251
views
Flag manifolds as incidence correspondences
Let $G$ be a reductive group, $B$ a Borel and $P_j$ the maximal parabolics, indexed by the vertices $j$ of the Dynkin diagram. Then $B = \bigcap_j P_j$, so the flag manifold $G/B$ injects into $\...
- 149k
4
votes
1
answer
201
views
Dimensions of $E_{7\frac{1}{2}}$
Is there much known about the dimensions $D$ of $E_{7\frac{1}{2}}$ (that is: $D_6.H_{32}$) beyond
$$
44\otimes44(def)=1\oplus945\oplus99(adj)\oplus891\, ?
$$
Generally, does a weight indexing scheme ...
- 4,631
11
votes
0
answers
582
views
The Grassmannian Gr(2,8) and an E7 surprise
Are there any mathematical explanations for the following surprising facts?
$$\int_{Gr(2,8)} c_{\text{top}}(TX(-2)) = 6556 = \frac{1}{2} \deg(E_7/P(\alpha_7)) + 1,$$
and
$$\int_{Gr(2,6)} c_{\text{top}}...
- 1,288
7
votes
0
answers
217
views
Does Deligne's exceptional series lead to an "exceptional K-theory"?
To a certain extent, Deligne's exceptional series $A_1 \subset A_2 \subset G_2 \subset D_4 \subset F_4 \subset E_6 \subset E_7 \subset E_8$ plays a role analogous to the classical series $A_n \subset ...
- 52.2k
6
votes
1
answer
382
views
Adjoint orbits of a finite group of type $G_2$
Let $q=p^\alpha$ be a prime power and $k=\mathbb{F}_q$. Let $G\subseteq \mathrm{GL}_N(k)$ be a simple finite group of Lie type, with root system of type $G_2$, and let $\mathfrak{g}\subseteq \mathfrak{...
- 993
7
votes
0
answers
205
views
Freudenthal geometries for exceptional simple Lie groups
When reading answer to this question I recall Freudenthal, Lie groups and foundations of geometry, 1964. In chapter 4 he describes 2-dim elliptic geometry, 2-dim projective geometry, 5-dim symplectic ...
user21230
9
votes
1
answer
371
views
Concrete description of an exceptional minuscule variety
Let $G$ be a complete reductive Lie group. A simple root $\alpha$ is said to be minuscule if the multiplicity of the coroot $\alpha^\vee$ in $\beta^\vee$ is at most $1$ for all positive roots $\beta$. ...
- 1,785
10
votes
1
answer
465
views
The double cover of $[W(E_7),W(E_7)] \cong Sp_6(\mathbb F_2)$ as a Galois group over $\mathbb Q$
I came across the following problem when I was trying to construct a certain type of homomorphisms from $\Gamma_{\mathbb Q}$ to $E^{sc}_7(\mathbb F_p)$ for any prime $p$:
Is the double cover of $Sp_6(...
- 433
15
votes
2
answers
400
views
Constructing $E_8$ from its branching to $A_8$
Background/motivation: One of the usual constructions of [the adjoint representation of] the $E_8$ exceptional Lie group (found, e.g., in J. F. Adams's, "Lectures on Exceptional Lie Groups", esp. chap....
- 28.2k
6
votes
2
answers
487
views
Constructing real forms of the Tits-Freudenthal magic square for (Rosenfeld) projective planes
If $\mathbb{K},\mathbb{L} \in \{\mathbb{R},\mathbb{C},\mathbb{H},\mathbb{O}\}$ then the Rosenfeld projective ("elliptic"?) plane $\mathbb{P}^2(\mathbb{K}\otimes\mathbb{L})$ is "the" compact Riemannian ...
- 28.2k
4
votes
1
answer
208
views
A question on complex semisimple Lie groups and $(\mathbb{C}^2, \omega)$
Consider $(\mathbb{C}^2, \omega)$ where $\omega$ is a non-degenerate complex skew-symmetric bilinear form on $\mathbb{C}^2$. Let us write
$V = (\mathbb{C}^2, \omega)$
There are many spaces one can ...
- 4,697
12
votes
2
answers
1k
views
Where can I find details of Elie Cartan's thesis?
I am interested in the details of Elie Cartan's thesis, and, more specifically the explicit construction of the exceptional Lie groups as groups of symmetries of some specific homogeneous polynomials (...
- 4,697
0
votes
1
answer
119
views
What is the stabilizer of the following $7$-dimensional cross-product?
Upon visiting Prof. Nurowski's homepage (http://www.fuw.edu.pl/~nurowski/), at the top of the page, there is the following $7$-dimensional cross product:
$e_1 e_2 = e_4$, $e_2 e_3 = e_5$,...
and so ...
- 4,697
5
votes
1
answer
292
views
What is the largest subgroup of $GL^{+}(7,\mathbb{R})$ which smoothly retracts onto $G_2$?
There is a nice smooth retraction from $\operatorname{GL}(n,\mathbb{C})$ onto $\operatorname{U}(n)$, which can be explained using polar decomposition. There is an analogous one from $\operatorname{GL}(...
- 4,697
4
votes
0
answers
138
views
Exceptional symmetric spaces with quaternionic structure
Following this and this question I found following chain of exceptional symmetric spaces being quaternionic manifolds. I listed dimensions as superscripts for reader convenience.
$F_{I}^{28}\subset ...
user21230
4
votes
0
answers
182
views
A few questions about $E_7$ and its symmetric spaces
My question about $E_6$ survived, so I post next episode. From the Yokota book I found out that there is $-1$ in $E_7$ Lie group. This book defines Lie group $E_7$ using 56-dimensional Freudenthal ...
user21230
7
votes
0
answers
578
views
A few questions about $E_6$ and its symmetric spaces
Preface
The purpose of my question - on high level - is to understand exceptional symmetric spaces. My latest idea is to embed them into Lie group. There is quite nice embedding of 32-dimensional $E_{...
user21230
6
votes
2
answers
109
views
Describing the action of $^2E_6(q)$
One of the constructions of the group $^2E_6(q)$ was presented by Tits in his paper "Les «formes réelles» des groupes de type $E_6$". It is being constructed by looking at the action of $^2E_6(q)$ on ...
- 161
1
vote
1
answer
287
views
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 ...
- 20.5k
2
votes
1
answer
284
views
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)...
- 20.5k
3
votes
1
answer
117
views
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 ...
- 20.5k
6
votes
0
answers
258
views
Exceptional symmetric spaces embedded in exceptional Lie group
In Yokota (1959) and Atsuyama (1977) papers one can find embedding of projective space $\mathbb OP^2$ into Lie group $F_4$. Lately I come to following idea to have embedding of all four projective ...
user21230
3
votes
1
answer
131
views
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 ...
- 2,061
3
votes
0
answers
165
views
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 ...
- 2,061
7
votes
2
answers
478
views
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 ...
- 2,975
1
vote
0
answers
181
views
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.
...
- 20.5k
2
votes
1
answer
307
views
$Spin(7)$ 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 $SO(7)<SO(8)$ ...
- 2,061