# Questions tagged [root-systems]

The root-systems tag has no usage guidance.

247
questions

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### Lie algebras, root systems and qubits

This post is about some concepts I am experimenting with. They are related to the Atiyah problem on configurations. They kind of mix Lie algebras and qubits. Given a compact (say semisimple) Lie group ...

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### Numerical method for mixed system of equations and nonlinear inequalities

I am currently encountering challenges in determining the solution method for the following system of equations and inequalities:
$$
\begin{aligned}
&F(x) = 0\\
&G(x) < 0\\
\end{aligned}
$$
...

1
vote

0
answers

38
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### Ultra-operations numbers (polynomials) [closed]

After Bring's root article, I became interested in understanding the theory of ultra numbers and their operations. There are very few vague concepts about these numbers on the Internet. I would be ...

0
votes

1
answer

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### Calculating relative root systems

Let $\mathbf{G}$ be a connected semisimple algebraic group defined over a field $k$. Let $T$ be a maximal torus of $\mathbf{G}$ defined over $k$, and let $S \subset T$ be a maximal $k$-split torus. ...

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1
answer

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### Linear independence of reciprocals of products of closed sets of roots in type $A$ inversion sets

Consider the root system $R$ for a Coxeter system $(W,S)$ of type $A_n$ with a choice of simple roots. Denote by $I(w)$ for $w\in W$ the set of positive roots $\beta\in R^+$ such that $w(\beta)$ is a ...

3
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2
answers

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### Describing characters of a reductive group in terms of characters of a maximal torus

Say I have a reductive complex algebraic group $G$ with maximal torus $T$ and associated Weyl group $W$. I would like to be able to say that the characters of $G$ are in bijection with the $W$-...

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104
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### Weyl group action on the Lie algebra [duplicate]

Let $W$ be the Weyl group of a complex semisimple Lie algebra $\frak{g}$. Certainly $W$ acts on the root system $R$ of $\frak{g}$ but can it be made to act on $\frak{g}$ or on the universal enveloping ...

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### Root space inner products and the partial order on roots

For a root system $R$ and a choice of positive roots $R^+$ it is a standard fact (see, e.g., Bourbaki, "Lie Groups and Lie Algebras," Theorem 1 of Section 1.3 of Chapter VI) that
if $(\...

7
votes

1
answer

319
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### 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 &...

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### Is this construction related to the geometric Langlands program perhaps?

Given a complex Lie algebra $\mathfrak{g}$, a choice of Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ and a dominant integral weight $\lambda$ of $\mathfrak{g}$, there is a natural construction ...

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### The Cartan is a complex vector space but the root system is real?

Let $\frak{g}$ be a complex semisimple Lie algebra with some choice of Cartan subalgebra $\frak{h}$. The dual space $\frak{h}^* = \mathrm{Hom}_{\mathbb{C}}(\frak{h},\mathbb{C})$ is a complex vector ...

2
votes

2
answers

261
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### Pairing a root with the half-sum of positive roots

Let $\frak{g}$ be a finite-dimensional complex simple Lie algebra together with a choice of Cartan subalgebra and associated root system $(\Delta, (-,-))$. Also we denote the half-sum of positive ...

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### Disconnected reductive algebraic groups in Sage

All simply connected split simple groups have been implement on Sage and it is possible to find their highest roots, fundamental weights, Dynkin diagrams or compute the tensor of two of their ...

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120
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### 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 ...

1
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1
answer

90
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### Problem in understanding a fact about Belavin-Drinfeld triple

A Belavin-Drinfeld triple associated to a simple Lie algebra $L$ is a triple $(\Gamma_1, \Gamma_2, \tau)$ where $\Gamma_1, \Gamma_2 \subseteq \Gamma$ ($\Gamma$ is a set of simple roots or fundamental ...

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1
answer

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### Sub-coroot lattices

[This is a sequel to the previous question sub-coroot systems, that has been answered! :-) ]
Let $T$ be a maximal torus of a compact Lie group $K$,
and let $\Lambda \subset {\mathfrak t}$ be the ...

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votes

1
answer

140
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### 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 $...

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0
answers

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### Multiplicative invariants of non-reduced root systems

It is a well known fact (cf. [1] VI.3.4 Thm. 1) that if $\Phi$ is a (reduced) root system with weight lattice $P$ and $W$ is the Weyl group of this root system, then the algebra of invariant ...

2
votes

1
answer

172
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### Find an analogue of Weyl chamber structure

Let $G$ be a split reductive group and let $T$ be a split maximal torus whose rank is $l$. Is it possible to find a base $\gamma_1,..., \gamma_l$ of the weight lattice $X(T)$ such that the cone $C$ in ...

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1
answer

77
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### Real roots along root strings

Let $A$ be a Cartan matrix, i.e. a $n\times n$ matrix with integer entries such that $A_{ii}=2$ and $A_{ij}\leq0$ for $i\neq j$. Then the corresponding Kac-Moody Lie algebra has a Cartan subalgebra $\...

2
votes

1
answer

244
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### Tensor product of fundamental representations

Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $V_1,\cdots, V_n$ be the fundamental representations (the irreducible ones with fundamental weights $\omega_1,\cdots,\omega_n$). Take a $k$-...

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votes

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92
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### finding positive roots for a polynomial [closed]

I have a polynomial, and I want to get the conditions for the number of positive roots
What are the different methods out there to determine these conditions?
this is the polynomial:
f(g)=A1g^5 + A2g^...

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### Schur polynomials are polynomials but also sequences on a lattice?

Monomial symmetric polynomials in $n$ variables $x_1, \ldots x_n$ form a natural basis for the space $\mathcal{S}_n$ of symmetric polynomials in $n$ variables and are defined by additive ...

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answer

254
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### Non-standard partial orders on root systems

Let $\frak{g}$ be a semisimple complex Lie algebra and let $\Delta$ be its associated root system with $\{\alpha_1, \dotsc, \alpha_l\}$ a choice of positive roots. As we all know - $\Delta$ admits a ...

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votes

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### A construction of Weyl-equivariant maps from the space of regular Cartan triples to the space of tuples of complex polynomials (up to scalar factors)

Let $G$ be a compact semisimple Lie group and let $T$ be a maximal torus in $G$. On the Lie algebra level, we have a real Lie algebra $\mathfrak{g}$ and a (particular) real slice, say $\mathfrak{t}$, ...

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### Intersection of certain subsets in a split connected reductive group $G$ related to affine open cover of $G/B$

Let $k$ be a field of characteristic zero and $G$ a split connected reductive group over $k$. Moreover, let $T$ be a split maximal torus of $G$ and $B\supset T$ a Borel subgroup. Additionally, we ...

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votes

1
answer

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### Why is the fundamental group of $\mathsf E_n$ cyclic of order $9 - n$?

Several years ago, I mentioned offhandedly to a colleague that I had noticed that, if you extend the $\mathsf E_n$ series downwards in the natural way, by removing nodes from the long arm of $\mathsf ...

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469
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### Typos in Bourbaki's root-system tables

A while ago, a colleague told me that he thought he remembered that there were typos in Bourbaki's tables in the English translation of "Groupes et algèbres de Lie", but that he could no ...

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139
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### Root systems and subroot systems

Given the root system $E_{6}$ with basis $\alpha_{1},\dotsc,\alpha_{6}$. How would I find all subroot systems of $E_{6}$ (up to Weyl equivalence) where I can write the basis of each subroot system in ...

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votes

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### The Weyl dimension formula for fundamental weights

The Weyl dimension formula is an equation to calculate the dimension of a simple $\frak{g}$-module $V_{\lambda}$, of highest weight $\lambda$, for $\frak{g}$ a complex semisimple Lie algebra. ...

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votes

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### Interpretation of the coefficients in the sum of positive roots

Take a finite Cartan datum with index set $I$, simple roots $\{\alpha_i\mid i \in I\}$ and positive roots $\Phi^+$. Let $2\rho=\sum_{\alpha\in\Phi^+}\alpha$ be the sum of the positive roots and write $...

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### PBW basis for the quantized enveloping Lie algebra of $\mathfrak{g}_2$

I would like to know if you have any reference where I can find the canonical PBW basis for $U_q(\mathfrak{g}_2),$ computed using the action of the braid group as defined by Luzstig.
Alternatively I ...

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votes

1
answer

183
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### Order ideals of positive root systems and avoiding group elements in the Weyl group

Let $X$ be the poset of positive roots of a finite root system of Dynkin type $Q$.
Question 1: In Dynkin type $A_n$, is it true that the poset of order ideals of $X$ is isomorphic to the poset of [2,...

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### A Lie-theoretic question regarding $B\ltimes \mathfrak{g}/\mathfrak{b}$

I am stuck on a seeming elementary Lie-theoretic question arising from a study of components of affine Springer fibers. Will be very grateful if somebody would like to share some insight, or ...

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vote

1
answer

294
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 ...

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views

### How to determine sublattices S of a root lattice R

Let $R$ be a root lattice of a irreducible root system $\Phi$.
Suppose $W$ is a Weyl group of $\Phi$ and $S$ is a sublattice of $R$ which is $W$-stable and satisfies $|R/S|<\infty$.
For example, ...

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vote

1
answer

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### A nice/simple relationship between the Chevalley generators of $\mathfrak{sp}_n$ and the Chevally generators of $\mathfrak{sl}_n$?

The Lie algebra $\mathfrak{sl}_n$ is defined to be the trace free matrices in $M_n(\mathbb{C})$. The Lie algebra $\mathfrak{so}_n$ is defined to be the matrices $A$ in $M_n(\mathbb{R})$ satisfying $A +...

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votes

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### Structure of the permutation groups acting on the root systems of Niemeier lattices of type $A_{k}^n$

I have been doing research on the Niemeier lattices with root systems of type, $A_{k}^n$ and I am particularly interested in the finite groups permuting the constituent root systems. These groups ...

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vote

0
answers

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### Condition for a sum of images of fundamental dominant weights to lie on a wall

Let $\Delta$ be a system of simple roots in a root system with Weyl group $W$. For $\alpha\in\Delta$, let $\varpi_\alpha$ be the corresponding fundamental dominant weight. Let $w\neq r$ be elements of ...

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### Why is the $A$-series root system best written in a vector space of one dimension higher?

In the classification of root systems, we have four families $A_n,B_n,C_n$, and $D_n$, and six exceptionals $E_6,E_7,E_8, F_4$, and $G_2$. For every non-exceptional case except $A_n$, the root system ...

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answer

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### Conjugation of root subgroups by the Weyl group

Fix a field $k$ of characteristic zero, and let $G$ be a connected reductive algebraic $k$-group of isotropic rank $\ge 1$. Fix a maximal $k$-split torus $S$, and let $\Phi_k$ be the relative root ...

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answers

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### Errata in N. A'Campo's "Tresses, monodromie et le groupe symplectique"

There are many small mistakes in this article. A great amount of them are concentrated in Lemma 2.
The setup for this lemma is the following. Let $R$ be a commutative ring and $n=2g+1$ or $n=2g$ a ...

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votes

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answer

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### A weight generalization of root systems?

For any simple complex Lie algebra $\frak{g}$, with a given choice of Cartan subalgebra $\frak{h}$, we have an associated root system $R \subseteq \frak{h}^*$. The properties of $R$ can be formalized ...

2
votes

1
answer

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### Action of the negative Cartan-Weyl generators on a highest weight element

Let $\frak{g}$ be a complex simple Lie algebra of rank $l$. For $\frak{h}$ a choice of Cartan subalgebra, let $\alpha_1, \cdots, \alpha_r$ be the corresponding choice of simple roots, $X_{\alpha_i}, ...

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votes

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answers

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### The group of fixed points of an involution of a Weyl group

Let $R$ be a reduced root system in a vector space $V$ over $\mathbb Q$.
Let $W=W(R)$ denote its Weyl group.
Let $S\subset R$ be a basis of $R$ (a system of simple roots).
Let $D=D(R,S)$ denote the ...

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0
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### Automorphism groups of which lattices act irreducibly on the ambient Euclidean space

(I asked this question on MSE a few days ago but it hasn't drawn any response yet.)
Let $V$ be a finite-dimensional real inner product space and let $L \subset V$ be a lattice of full rank. Consider ...

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votes

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### Existence and uniqueness of solution of a nonlinear system

I need a proof of the following result to calculate a Nash equilibrium in the Showcase Showdown game.
For all $n>1$, the system of equations
$$\left\{
\begin{aligned}
(1+e^{x}(-1+x))^{n-2}&=\...

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votes

1
answer

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### Action of split torus on positive root spaces

Let $G$ be a connected reductive group over a field $k$ (not necessarily algebraically closed). Let $S$ be a maximal split torus in $G$ with relative root system $\Phi = \Phi_k(S,G)$. Let $\Phi^+$ ...

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### Which representations of $\mathfrak{sl}(2)$ are homomorphic images of the tensor product of finitely many copies of $\mathbb{C}^2$?

My questions may turn out to be related to Schur functors.
If $\mathfrak{g}$ is a complex semisimple Lie algebra and $\lambda$ is the highest weight of an irreducible representation $V$ of $\mathfrak{...

3
votes

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219
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### Singular del Pezzo surfaces and degeneration of root systems

Let $S$ be a smooth del Pezzo surface of degree $d$ and $K_S^*$ the anticanonical class. It is well known that the set of classes
$$R(S)=\{\alpha\in H^2(S,\mathbb Z)|\alpha^2=-2,\alpha\cdot K_S^*=0\},$...