18
votes
Accepted
Where does the really nice '8-dimensional' description of the $E_7$ root system come from?
I don't know who found this presentation first but I can imagine that already Cartan knew it since it comes from a symmetric space. More precisely, $\mathfrak g=E_7$ has an involution $\theta$ whose ...
- 15.5k
18
votes
Where do root systems arise in mathematics?
I first came across root systems in the classification of finite reflection groups. A point group $\Gamma\subseteq\mathrm O(\Bbb R^n)$ is a reflection group if it is generated by reflections at ...
15
votes
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Why, conceptually, does the torus normalizer in $G_2$ split?
Here's a description that doesn't use octonions; instead, it uses the definition of $\mathrm{G}_2$ as the stabilizer of a $3$-form on $\mathbb{R}^7$. For simplicity, I'll do this for the split-form, ...
- 108k
14
votes
Accepted
Are (semi)simple Lie groups some sort of "homotopy quotient groups" of their maximal tori?
Here's the "answer" that I started writing, then put away for a while. The short answer is: although "T//W" is not the same as G, they "look sort of the same" from the point of view of certain ...
- 27.2k
13
votes
Where do root systems arise in mathematics?
They arise in the representation theory of quivers: Gabriel's theorem says that a connected quiver has finite representation theory type if and only if it is of type ADE, and then the indecomposable ...
12
votes
Accepted
Typos in Bourbaki's root-system tables
Not speaking to the English edition, but:
I have (an electronic version of) the 1981 Masson edition (in French) of books IV-VI of the LIE volume, and that one contains a list of typos at the end. You ...
- 2,454
11
votes
Definition of $\textrm{GSpin}_{2n}$ and its root datum
A good concise reference is Deligne's article on the Weil conjectures for K3 surfaces. See "La conjecture de Weil pour les surfaces K3" in Inventiones 15 (1971/72): 206-226. An English version is ...
- 13.3k
11
votes
Accepted
Non-faithful irreducible representations of simple Lie groups
Let $G_{sc}$ be as in the answer by Victor Protsak and let $\varpi_1$, $\ldots$, $\varpi_l$ be the fundamental dominant weights.
Let $\lambda$ be a dominant weight and write $\lambda = \sum_{i = 1}^...
- 2,821
11
votes
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Generalized root systems and reflection groups
If we place no restrictions on $k$, then this is precisely the class of finite groups that are generated by involutions.
In particular, if $G$ is any finite group of order $n$, then in the left ...
- 8,483
10
votes
Accepted
Subtori of groups of type E6
This question is precisely answered by Borel–de Siebenthal theory. Ignoring rationality issues (i.e., base changing to an algebraic closure), and fundamental groups, the possible types of $H$ are $...
- 12.9k
10
votes
Where do root systems arise in mathematics?
The eigenvalue distribution functions of random matrices in different universality classes are determined by the multiplicities of the restricted roots of the corresponding symmetric spaces, see ...
10
votes
Why, conceptually, does the torus normalizer in $G_2$ split?
Fix an $\mathbb{H} \subset \mathbb{O}$. This choice selects an involution of $\mathbb{O}$ which is $+1$ on $\mathbb{H}$ and $-1$ on $\mathbb{H}^\perp$. The centralizer of this involution is an $SO(4)$,...
- 54.6k
9
votes
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Does the classification of reductive groups follow from that of semisimple groups?
As indicated in the comments, there is no need to redo the entire classification argument when passing from "semisimple" to "reductive". But it's useful to recall some of the history. The emphasis ...
- 52.9k
9
votes
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Del Pezzo surfaces and Picard-Lefschetz theory
Indeed you can see it this way. This is my symplectic geometer's
perspective on it (I blame Paul Seidel's Lecture notes on four-dimensional Dehn twists).
Consider the $n$-point blow-up of $\mathbf{CP}...
- 7,005
9
votes
Accepted
Number of real roots in type $\tilde{E}_8$
I don't believe there is a reference for this, for this follows immediately from the description of real roots in affine root systems. Namely, by Proposition 6.3(a) in "Infinite dimensional Lie ...
- 3,186
9
votes
A weight generalization of root systems?
If $\Phi$ is a(n abstract) root system in a Euclidean vector space $V$, then we have notions of simple roots $\alpha_1,\ldots,\alpha_r$, fundamental weights $\omega_1,\ldots,\omega_r$, root lattice $Q ...
- 24.2k
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
8
votes
Accepted
Non-reduced, non-crystallographic root systems
Let's stick to the OP's definition of a root system.
Let $\Phi_0$ be the set of normalized roots $\frac{\alpha}{||\alpha||}$, $\alpha\in\Phi$. This is a root system satisfying 1, 2 and 3. Thus it is ...
- 15.5k
8
votes
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Each $w\in W$ can be expressed as product of distinct reflections?
The answer is "yes" and there is a geometric explanation.
Let $\mathcal{H}$ denote the set of hyperplanes corresponding to the reflections $s_\alpha$ with $\alpha\in\Phi^+$ (note that $s_\alpha=s_{-\...
- 2,928
8
votes
Non-faithful irreducible representations of simple Lie groups
By the highest weight theory, finite-dimensional irreducible representations of a finite-dimensional simple Lie algebra $\mathfrak{g}$ are parametrized by the dominant weights $\lambda$ in the weight ...
- 14.5k
8
votes
Which representations of $\mathfrak{sl}(2)$ are homomorphic images of the tensor product of finitely many copies of $\mathbb{C}^2$?
$\def\CC{\mathbb{C}}\def\ZZ{\mathbb{Z}}\def\sl{\mathfrak{sl}}$In the comments to hm2020's answer, the OP explains that they want a surjection from $(\mathbb{C}^2)^{\otimes m}$, not from a direct sum $\...
- 156k
8
votes
Typos in Bourbaki's root-system tables
There is a typo that I found last year: on page 216 "Système de type $E_{7}$", the second fundamental weight $\varpi_{2}$ should be
$$\frac{1}{2}(4\alpha_{1}+7\alpha_{2}+8\alpha_{3}+12\...
- 391
8
votes
Accepted
Why is the fundamental group of $\mathsf E_n$ cyclic of order $9 - n$?
If we define the $E_n$ lattice in the algebraic geometer's way as the orthogonal complement of $(1,\dots, 1, 3)$ in the Lorentzian unimodular lattice $I_{n,1}$, i.e. in $\mathbb Z^{n+1}$ with ...
- 148k
8
votes
Non-standard partial orders on root systems
Another partial order which comes up a lot is to take the transitive closure of the relation that, if $\langle \alpha_i, \beta \rangle < 0$, then $\beta \prec s_i(\beta)$.
For example, consider $...
- 156k
8
votes
Accepted
Multiplication factors in folding root systems and Lie algebras by automorphisms
Suppose we fold a root system $(\Phi,\Delta)$ to a root system $(\Phi^\sigma,\Delta^\sigma)$. There exists two conventions:
long roots of $(\Phi^\sigma,\Delta^\sigma)$ correspond to multiple roots in ...
- 3,054
7
votes
Where does the really nice '8-dimensional' description of the $E_7$ root system come from?
[edited to include more examples of the construction and fix a typo]
This can also be seen purely in terms of lattices or quadratic forms.
As you note, for any $n$ the vectors in the slice $\sum_{i=0}...
- 79.9k
7
votes
Accepted
Why is the root poset is graded by height?
Let $(\cdot,\cdot)$ be a positve definite Weyl group invariant product on $\mathbb{R}\Phi$. Let $\beta$ and $\gamma$ be positive roots with $\beta\leq \gamma$ and $h(\gamma)-h(\beta)\geq 2$. Let $v=\...
- 8,866
7
votes
Accepted
Existence of a weight of a representation in the fundamental Weyl chamber
Here's maybe another (more conceptual?) way to think about it.
First of all, if $\mu_1$ is a dominant weight which appears with nonzero multiplicty in $V^{\lambda_1}$, and $\mu_2$ is a dominant ...
- 24.2k
7
votes
Accepted
Root lattices and (resolutions of) singular cubic surfaces
I think you will find everything you want in Demazure, Pinkham, Teissier (eds.), Séminaire sur les singularités des surfaces, Springer LNM 777. Over non-closed fields there's also an article by Coray ...
- 4,185
7
votes
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Lattice structure in the root poset
The root poset for $\tilde{A_2}$ is shown in Figure 4.5 in the same reference and copied below. One can check that the elements labelled $112$ and $221$ have both $100$ and $010$ as maximal common ...
- 2,743
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