20 votes
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Is the assignment of a root system to a complex semisimple Lie algebra functorial?

For everything you're requesting, it seems reasonable only to consider this question using maps that are isomorphisms, since: (i) the zero map between semisimple Lie algebras has no reasonable "...
17 votes
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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 ...
Friedrich Knop's user avatar
14 votes
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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 ...
Charles Rezk's user avatar
  • 26.6k
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 ...
Jim Belk's user avatar
  • 8,433
11 votes
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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 ...
Torsten Schoeneberg's user avatar
10 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 ...
Marty's user avatar
  • 13.1k
10 votes
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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 $...
LSpice's user avatar
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10 votes
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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}^...
spin's user avatar
  • 2,781
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}...
Jonny Evans's user avatar
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9 votes
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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 ...
Andrei Smolensky's user avatar
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 ...
David E Speyer's user avatar
8 votes
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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 ...
Friedrich Knop's user avatar
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_{-\...
Uriya First's user avatar
  • 2,846
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 $\...
David E Speyer's user avatar
8 votes
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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 ...
Will Sawin's user avatar
  • 135k
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 $...
David E Speyer's user avatar
7 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 ...
Jim Humphreys's user avatar
7 votes
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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=\...
Peter McNamara's user avatar
7 votes
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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 ...
Sam Hopkins's user avatar
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7 votes
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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 ...
Martin Bright's user avatar
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 ...
Christian Gaetz's user avatar
7 votes

Number of reduced decompositions of the longest element of the Weyl group

Around the time Fomin and I wrote this paper, Tao Kai Lam applied the technique to type $D_n$. It emerged that it was "natural" to weight a reduced decomposition $\rho$ by $2^{d(\rho)}$, ...
Richard Stanley's user avatar
7 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 ...
Sam Hopkins's user avatar
  • 22.7k
7 votes

Order ideals of positive root systems and avoiding group elements in the Weyl group

Regarding Q1: I believe this is proved in "A Distributive Lattice Structure Connecting Dyck Paths, Noncrossing Partitions and 312-avoiding Permutations" by Barcucci et al. https://doi.org/10....
Sam Hopkins's user avatar
  • 22.7k
7 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\...
Erica's user avatar
  • 286
6 votes
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Motivation behind Panyushev's "constant-averages-along-orbits" conjecture

In "The root poset and its relatives" (https://arxiv.org/abs/math/0502385) Panyushev established (see Corollary 3.4) that the average size of an antichain of the root poset $\Phi^+$ of an irreducible ...
Sam Hopkins's user avatar
  • 22.7k
6 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}...
Noam D. Elkies's user avatar
6 votes
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Difference of adjacent dominant weights is a root?

This is true. See "The partial order of dominant weights" by John Stembridge, 1998 (https://www.sciencedirect.com/science/article/pii/S0001870898917364). In particular look at Corollary 2.7. ...
Sam Hopkins's user avatar
  • 22.7k
6 votes

Fixed Points of the Weyl Group action on a Maximal Torus and the Center of a Reductive Group

Let $X = Hom(T, G_m)$ and $Y = Hom(G_m, T)$ be the character and cocharacter lattices of $T$, respectively. Let $k$ be the (algebraically closed) ground field. Note that $T = Spec(k[X])$ (a ...
Marty's user avatar
  • 13.1k
6 votes
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Reference for parabolic root systems

Try section "Relative roots" here: http://math.stanford.edu/~conrad/249BW16Page/handouts/249B_2016.pdf They originate from the paper of Borel and Tits, I guess: http://www.numdam.org/item/...
Victor Petrov's user avatar

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