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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7 votes
1 answer
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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 $\omega_i$ ...
7 votes
2 answers
398 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 ...
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11 votes
1 answer
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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)...
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4 votes
0 answers
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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 $\mathrm{Sp}_{56}(\mathbb R)$ ...
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2 votes
1 answer
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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 ...
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9 votes
1 answer
338 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 ...
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4 votes
0 answers
104 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, \, ...
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2 votes
0 answers
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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 ...
8 votes
1 answer
292 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 ...
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4 votes
0 answers
58 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?
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4 votes
2 answers
378 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 ...
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6 votes
2 answers
255 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
90 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 ...
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16 votes
0 answers
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$8 \times 31 = 8 \times 31$?

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}...
6 votes
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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 ...
22 votes
1 answer
520 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}...
8 votes
1 answer
213 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 ...
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5 votes
0 answers
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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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12 votes
1 answer
229 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 $\...
4 votes
1 answer
199 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 ...
11 votes
0 answers
567 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}}...
7 votes
0 answers
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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 ...
5 votes
1 answer
244 views

Adjoint orbits of a finite group of type $G_2$ [reference request]

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{...
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7 votes
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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 ...
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9 votes
1 answer
353 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 $\beta$. ...
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10 votes
1 answer
432 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(...
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14 votes
2 answers
374 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....
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5 votes
2 answers
449 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 ...
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4 votes
1 answer
200 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 ...
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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 (...
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0 votes
1 answer
118 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 ...
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5 votes
1 answer
288 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}(...
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4 votes
0 answers
132 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 ...
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4 votes
0 answers
174 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 ...
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7 votes
0 answers
549 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_{...
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6 votes
2 answers
106 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 ...
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1 vote
1 answer
268 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 ...
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2 votes
1 answer
263 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)...
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3 votes
1 answer
112 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 ...
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6 votes
0 answers
236 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 ...
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3 votes
1 answer
119 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 ...
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3 votes
0 answers
164 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 ...
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7 votes
2 answers
416 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 ...
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1 vote
0 answers
179 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. ...
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2 votes
1 answer
285 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 $SO(7)<SO(8)$ ...
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8 votes
1 answer
378 views

$Spin(7)$ 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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0 votes
2 answers
579 views

Decomposition of $S^7=\operatorname{Spin}(7)/G_2$

$\DeclareMathOperator\Spin{Spin}$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 ...
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9 votes
1 answer
472 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 ...
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1 vote
2 answers
462 views

smallest simplest $E_8$ -module

What is the smallest simplest(non-trivial) $E_8$ -module ?
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26 votes
4 answers
4k 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. $Spin(8)$ ...