Questions tagged [root-systems]

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Does -I belong to Weyl group?

Let $\Phi$ be an irreducible root system, with positive roots $\Phi^+$ relative to the base $\Delta$. If $W$ is the Weyl group, how can I determine if $-I$ belongs to $W$? Equivalently how can I see ...
1df5e76's user avatar
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6 votes
1 answer
1k views

Finite dimensional Lie algebra with non-degenerate invariant bilinear forms $\Omega_{ab}$

Firstly, my apology to MO experts that I am in a more science/physics background (a PhD). So please feel free refine/modify/comment my language if I have different math accents than yours. From ...
miss-tery's user avatar
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5 votes
3 answers
2k views

Complete classification of six dimensional non-semi simple Lie algebra

I would aim to know the complete classification of 6 dimensional non-semi simple Lie algebra (here the dimension stands for the generators; or the dimension $\leq 6$). In this paper, in page 7, it ...
cycles's user avatar
  • 81
4 votes
1 answer
379 views

The existence of a finite dimensional Lie algebra with a given symmetric invariant metric

The question is motivated by a more broad perspective in another MO post and here, but here we would like to understand a specific case (our question potentially connects to / is motivated b Quantum ...
miss-tery's user avatar
  • 745
3 votes
2 answers
341 views

Partial ordering on $\mathfrak{h}^*$ and Bruhat ordering

In section 5.2 (p.95) of Representations of Semisimple Lie Algebras in the BGG Category $\mathcal{O}$. Let $\mu\le \lambda$ if $\lambda-\mu\in \Gamma$, where $\Gamma$ is the set of $\mathbb{Z}^{\ge 0}$...
James Cheung's user avatar
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20 votes
1 answer
876 views

Curious fact about number of roots of $\mathfrak{sl}_n$

The Lie algebra $\mathfrak{sl}_n $ has many special features which are not shared by other simple Lie algebras, for example all of its fundamental representations are minuscule. I recently discovered ...
Joel Kamnitzer's user avatar
13 votes
3 answers
898 views

Where does the really nice '8-dimensional' description of the $E_7$ root system come from?

The Wikipedia page on $E_7$ tells me: Even though the roots span a 7-dimensional space, it is more symmetric and convenient to represent them as vectors lying in a 7-dimensional subspace of an 8-...
Vincent's user avatar
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10 votes
0 answers
397 views

Gram matrix determinant in dimension 4 and $E_8$

Consider a determinant of a Gram matrix in dimension $4$. $$\begin{vmatrix} 1 & -\cos(\alpha_1) & -\cos(\alpha_2) & -\cos(\alpha_3)\\ -\cos(\alpha_1) & 1 & -\cos(\alpha_6)& -\...
Daniil Rudenko's user avatar
9 votes
2 answers
572 views

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

Let $R$ be a reduced root system, $W$ the associated Weyl group, and $w_0 \in W$ the longest element of $W$. In general $w_0$ admits more than one reduced decomposition into a product of reflections, ...
Bas Winkelman's user avatar
9 votes
1 answer
925 views

Technical lemma on root systems, reduced to linear algebra

Update: I have posted the case of $G = SL(n)$ as a different question here. This is a technical lemma I am currently stuck at. Any suggestions about how to proceed are welcome. Let $G$ be a split ...
Abhishek Parab's user avatar
9 votes
1 answer
679 views

Del Pezzo surfaces and Picard-Lefschetz theory

Let $X$ be a smooth compact del Pezzo surface. For instance, one can consider the most classical case of a cubic surface. It is well known that the Picard lattice of $X$ is related to a root system (...
Daniil Rudenko's user avatar
8 votes
1 answer
372 views

Motivation behind Panyushev's "constant-averages-along-orbits" conjecture

In his article "On orbits of antichains of positive roots" (European Journal of Combinatorics 30 (2009) 586–594, Dmitri Panyushev discusses an interesting self-map on the set of antichains of a finite ...
James Propp's user avatar
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7 votes
2 answers
489 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 ...
jdc's user avatar
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6 votes
2 answers
1k views

Non-faithful irreducible representations of simple Lie groups

For a complex simple Lie algebra $\frak{g}$, which of its finite dimensional irreducible representations give non-faithful representations of the corresponding simply-connected compact Lie group. ...
Nadia SUSY's user avatar
6 votes
3 answers
735 views

Existence of a weight of a representation in the fundamental Weyl chamber

Let $\mathfrak g$ be a complex simple Lie algebra. Fix a Cartan subalgebra $\mathfrak h$ of $\mathfrak g$, let $\Delta$ denote the corresponding root system. Pick a partial order on $\mathfrak h$, ...
emiliocba's user avatar
  • 2,279
6 votes
1 answer
403 views

Dynkin diagram of the centralizer of a semisimple element in a Levi subgroup

Let $G$ be a connected reductive group over an algebraically closed field and consider a semisimple element $s \in G$ and let $L$ be a Levi subgroup containing $s$. My question is about the two ways ...
Matthias Klupsch's user avatar
6 votes
2 answers
436 views

Relation between linear and multiplicative invariants of Weyl groups

Let $\mathcal{W}$ be a Weyl group, acting variously on a root system $\Phi$, the real vector space $V = \langle\Phi\rangle_{\mathbb{R}}$ in which they live, or the associated weight lattice $P \subset ...
Gro-Tsen's user avatar
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6 votes
0 answers
325 views

Non-crystallographic cluster algebras

Background Fomin and Zelevinsky have introduced cluster algebras in an influential article. To define a cluster algebra, Fomin and Zelevinsky have defined a mutation of seeds. Here, a seed $(\mathbf{...
Philipp Lampe's user avatar
6 votes
3 answers
647 views

References about Hasse diagrams of root systems

This is to ask about references of Hasse diagrams of irreducible root systems. I found here and there nice pictures of root systems of type $E$. I would like to ask for Hasse diagrams of classical ...
user's user avatar
  • 287
5 votes
2 answers
491 views

Even unimodular lattices with root system $32 A_1$

I'm studying Venkov's proof of the classification of even unimodular rank 24 lattices, and it prompted the following question. For an even unimodular lattice $L$, let $R(L)= \{ x \in L : (x,x) =2\}$ ...
Ariyan Javanpeykar's user avatar
5 votes
2 answers
2k views

Which linear combinations of simple roots are roots

Let $\Delta$ be the root system of a complex simple Lie algebra, $\Delta^+$ be positive roots and $\Pi$ be simple roots. We view $\Pi$ as nodes of the Dynkin diagram. Then for any two simple roots $\...
Xin Nie's user avatar
  • 1,764
5 votes
2 answers
921 views

Root system automorphisms as inner automorphisms of extended Chevalley group

For each automorphism $\sigma$ of a root system $\Phi$ there is a unique automorphism of the Chevalley group $G(\Phi,R)$ such that $\sigma(x_\alpha(t))=x_{\sigma\alpha}(t')$. While conjugating by ...
Andrei Smolensky's user avatar
5 votes
1 answer
185 views

Number of real roots in type $\tilde{E}_8$

Let $\Phi_+$ be the set of all positive roots for a Kac-Moody algebra. Denote by $\alpha_i$ the simple root associated with node $i$ by for $i \in \{1, \ldots, n-1\}$ and by $\beta$ the simple root ...
Jianrong Li's user avatar
  • 6,101
5 votes
2 answers
570 views

$\textrm{GSp}_{4}^{\wedge} \cong \textrm{GSp}_4$

Everything is over an algebraically closed field of characteristic $\neq 2$. I had constructed a root datum for $\textrm{GSp}_4$ in a less than ideal way, and I had hoped to show that it was self ...
D_S's user avatar
  • 6,100
5 votes
2 answers
629 views

How to find faces of polytope defined by a Weyl orbit

A few days ago I asked the following question at MSE and received no answer. I thought I would try here. Let $\xi$ be an integral dominant weight of an irreducible root system $\Delta$, and let $\...
James Mracek's user avatar
4 votes
1 answer
177 views

Can we have a nontrivial division of a irreducible root system as the union of two closed sub-root systems?

The question is related to this MO question. Let $(\Phi, E)$ be a irreducible crystallographic root system where $\Phi$ is the set of all roots and $E$ is the $\mathbb{R}$-span of $\Phi$. As in the ...
Zhaoting Wei's user avatar
  • 8,657
4 votes
0 answers
128 views

Panyushev's conjectured duality for root poset antichains

In his 2004 paper "ad-nilpotent ideals of a Borel subalgebra: generators and duality" (https://www.sciencedirect.com/science/article/pii/S0021869303006380), Panyushev conjectured (Conjecture 6.1) the ...
Sam Hopkins's user avatar
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4 votes
1 answer
275 views

The image of a base of absolute roots is a base of relative roots

$G$ is a connected, reductive group over a field $k$, $S$ is a maximal $k$-split torus of $G$, and $_k \Phi = \Phi(S,G)$ is the set of roots of $S$ in $G$. Equivalently, $_k \Phi$ is the restriction ...
D_S's user avatar
  • 6,100
3 votes
1 answer
149 views

Sub-coroot systems

Let $T$ be a maximal torus of a compact Lie group $K$, and let $\Psi \subset {\mathfrak t}$ be the (finite) set of coroots for $(K,T)$, where $\mathfrak t$ is the Lie algebra of $T$. Assume now that $...
bernardorim's user avatar
1 vote
0 answers
107 views

Nontrivial relations of the irreducible root systems

For the root system of the type $A_n$, the roots are $\alpha _{i,j}$, $1\le i\neq j\le n$, we have the nontrivial relations $(x_{i,j} (t), x_{j,k}(u)) = x_{i,k}(tu)$ if $i, j, k$ are distinct. ($x_{i,...
Ami's user avatar
  • 332
1 vote
1 answer
227 views

Can we have a nontrivial division of a irreducible root system as $\Phi=\Phi_{[\lambda]}\cup \Phi_{[\mu]}$?

Let $(\mathfrak{g},\mathfrak{h},\Phi)$ be a root system of a complex simple Lie algebra, where $\Phi$ is the set of all roots. For each $\alpha\in \Phi$, let $\alpha^{\vee}=2\alpha/(\alpha,\alpha)$ be ...
Zhaoting Wei's user avatar
  • 8,657
1 vote
1 answer
334 views

The principal congruence subgroup of the symplectic group over the integers

Consider the symplectic group $\text{Sp}_{2g}(\mathbb{Z})$ over the integers. It has a classical root system $C_g$ and associated root subgroups $U_\varphi$ for $\varphi\in C_g$. These subgroups are ...
user avatar
-7 votes
4 answers
988 views

$E_6$, $E_8$, and Coxeter's (anti-)prismatic projections of the n-dimensional cross-polytopes

Edited 1/21/2018 to add the following: Here is a DropBox link https://www.dropbox.com/s/7rtt0iqmgimsgzu/Zumkeller_edge-magic.pdf?dl=0 to a PDF showing how my team used biomolecular first ...
David Halitsky's user avatar