Questions tagged [root-systems]

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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 ...
Malkoun's user avatar
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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: $$ \begin{aligned} &F(x) = 0\\ &G(x) < 0\\ \end{aligned} $$ ...
AnNam's user avatar
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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 ...
Aleksandr's user avatar
0 votes
1 answer
103 views

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. ...
Ann's user avatar
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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 ...
Matt Samuel's user avatar
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3 votes
2 answers
129 views

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 $W$-...
Henry Talbott's user avatar
1 vote
0 answers
104 views

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 ...
Lorenzo Del Vecchiopontopolos's user avatar
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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 $(\...
Fantas Anadolou's user avatar
7 votes
1 answer
319 views

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 &...
Марина Marina S's user avatar
7 votes
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649 views

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 ...
Malkoun's user avatar
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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 ...
Jake Wetlock's user avatar
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2 votes
2 answers
261 views

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 ...
Didier de Montblazon's user avatar
3 votes
0 answers
141 views

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 ...
dm82424's user avatar
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1 answer
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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 ...
emiliocba's user avatar
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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 ...
Anil Bagchi.'s user avatar
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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 ...
bernardorim's user avatar
3 votes
1 answer
140 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
2 votes
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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 ...
G. Gallego's user avatar
2 votes
1 answer
172 views

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 ...
Allen Lee's user avatar
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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 $\...
freeRmodule's user avatar
2 votes
1 answer
244 views

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 $k$-...
Jun Yang's user avatar
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1 answer
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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^...
Doaa mahmoud's user avatar
4 votes
0 answers
178 views

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 ...
Arnold Mckenzie's user avatar
5 votes
1 answer
254 views

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 ...
Didier de Montblazon's user avatar
2 votes
0 answers
46 views

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 ...
KKD's user avatar
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7 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 ...
LSpice's user avatar
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16 votes
2 answers
469 views

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 ...
LSpice's user avatar
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2 votes
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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 ...
PSHINH2's user avatar
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2 votes
0 answers
226 views

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. ...
Dave Shulman's user avatar
4 votes
0 answers
88 views

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 $...
Andrew's user avatar
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5 votes
1 answer
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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 ...
Alessandro Carotenuto's user avatar
6 votes
1 answer
183 views

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,...
Mare's user avatar
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9 votes
0 answers
257 views

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 ...
Cheng-Chiang Tsai's user avatar
1 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 ...
user avatar
3 votes
0 answers
81 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, ...
Fuutorider's user avatar
1 vote
1 answer
200 views

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 +...
Jake Wetlock's user avatar
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4 votes
1 answer
170 views

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 ...
Sean Miller's user avatar
1 vote
0 answers
221 views

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 ...
Stefan  Dawydiak's user avatar
4 votes
0 answers
96 views

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 ...
johhnyelgerton's user avatar
3 votes
1 answer
344 views

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 ...
Joshua Ruiter's user avatar
2 votes
0 answers
187 views

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 ...
user avatar
4 votes
1 answer
189 views

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 ...
johhnyelgerton's user avatar
2 votes
1 answer
213 views

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}, ...
johhnyelgerton's user avatar
3 votes
0 answers
144 views

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 ...
Mikhail Borovoi's user avatar
1 vote
0 answers
59 views

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 ...
Ivan Solonenko's user avatar
2 votes
0 answers
119 views

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}&=\...
José María Grau Ribas's user avatar
3 votes
1 answer
136 views

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 $\Phi^+$ ...
Laura's user avatar
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9 votes
2 answers
1k views

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{...
Malkoun's user avatar
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3 votes
0 answers
219 views

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\},$...
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