Questions tagged [root-systems]

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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 ...
Joshua Ruiter's user avatar
2 votes
0 answers
189 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
210 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
222 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
169 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
65 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
120 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
142 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
  • 33
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
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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\},$...
AG learner's user avatar
  • 1,701
2 votes
1 answer
139 views

Algorithm to determine if a vector in the geometric representation of a Coxeter group is proportional to a root

Let $W$ be a Coxeter group, and let $V$ be its geometric representation (as defined for instance in Section 5.3 of Humphreys' book Reflection groups and Coxeter groups). Let $v\in V\backslash\{0\}$ (...
Thomas Gobet's user avatar
3 votes
0 answers
142 views

Relation between root subgroups and the root system in unitary groups

Consider a 4-dimensional non-degenerate unitary space over a field of order 4. It can be shown that there are 45 isotropic lines. For each such a line one can associate a unitary transvection and each ...
Alexey Staroletov's user avatar
3 votes
1 answer
550 views

A formula for the dual Coxeter number

Let $\Phi$ be the root system of a finite dimensional simple Lie algebra $\mathfrak g$, with dual Coxeter number $h^\vee$. Let $\alpha_0\in \Phi$ be a long root (if all the roots have the same length, ...
André Henriques's user avatar
2 votes
0 answers
152 views

What is $f^*TX$ for a general morphism $f\colon\mathbb{P}^1\to X$?

Let $X$ be a projective homogeneous space over $\mathbb{C}$, i.e. $G/P$ where $G$ is a simple, simply connected linear algebraic group and $P$ is a parabolic subgroup. Let $f\colon\mathbb{P}^1\to X$ ...
Christoph Mark's user avatar
4 votes
0 answers
237 views

Eigenvalues and eigenvectors of the exceptional simple Lie group E6, E7, E8

What is the significance of the eigenvalues and eigenvectors of the exceptional simple Lie group root lattice to the Lie group or other mathematics branches? For example, E6, we have $$ \left( \begin{...
annie marie cœur's user avatar
1 vote
1 answer
61 views

How to verify that an element in the root lattice is an imaginary root of a non-hyperbolic root system?

In my research I encounter some elements in a root lattice and I would like to verify that these elements are imaginary roots. Consider the root system $J_{6, 11}$ with the following Dynkin diagram: \...
Jianrong Li's user avatar
  • 6,101
7 votes
0 answers
332 views

Why are fundamental weights denoted by omega?

In my field (and many others, I believe) the absolutely standard notation for the fundamental weights of a root system is lowercase omega: $\omega$. Recently I was taken aback to receive a copyedited ...
Igor Makhlin's user avatar
  • 3,493
9 votes
1 answer
294 views

Action of Weyl group on regions of Shi arrangement

This is an elaboration of a question which was aked on MO several years ago, which was unanswered but deleted by the question-asker. I hope it is okay to elaborate on a deleted question like this; for ...
Sam Hopkins's user avatar
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9 votes
2 answers
575 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
1 vote
0 answers
122 views

Uniqueness in Mare combinatorics and bounds on Gromov-Witten invariants

Let $R$ be the root system of a Weyl group $W$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. the set of positive roots $\alpha$ such that $\ell(s_\alpha)=2\operatorname{ht}\alpha-1$. Based ...
Christoph Mark's user avatar
0 votes
0 answers
114 views

What is the name of the following root system?

The Dynkin diagram of the root system of affine $D_4$ is $$ \circ \quad \circ \quad \circ \quad \circ \\ \circ $$ where all of the four vertices in the first row connects to the vertex in the second ...
Jianrong Li's user avatar
  • 6,101
7 votes
0 answers
174 views

Regarding $F_4$ and $G_2$ Lie algebras, do there exist $F_n$ or $G_n$ families of Lie algebras?

For example, $E_6$ exceptional Lie algebra is part of the $E_n$ series of Lie algebras (Kac-Moody algebras). Are $F_4$ or $G_2$ maybe also parts of some $F_n$ or $G_n$ series of Lie algebras or are ...
Domagoj Hranjec's user avatar
3 votes
1 answer
207 views

when a set of roots extend to a system of simple roots

Given a set of roots in a root system, assume that the pairing of each two roots in this set is not positive. Then clearly the set gives a closed root subsystem. My question is, when this set extends ...
JJH's user avatar
  • 1,447
12 votes
2 answers
470 views

Lattice structure in the root poset

Let $W$ be a Coxeter group with simple generators $s_1$, $s_2$, ..., $s_r$. Let $\Phi^+$ be the corresponding positive root system, with $\alpha_i$ the positive root corresponding to $s_i$. Bjorner ...
David E Speyer's user avatar
4 votes
1 answer
234 views

A transversal for the $\operatorname{Ad}(K)$ action on a sphere in $\mathfrak{p}$

This exercise level question has been unanswered on MSE for a few years. I hope you can answer it either there or here. $G$ is a semisimple Lie group with a choice of Cartan decomposition on its Lie ...
Chertopkhanov on Malek Adel's user avatar
6 votes
1 answer
451 views

Is the connected centralizer of a semisimple element in a connected reductive group also a centralizer?

Let $G$ be a connected reductive algebraic group defined over an algebraically closed field and let $g\in G$ be semisimple. Write $C=\mathrm{C}_G(g)$ and $C^\circ=\mathrm{C}_G(g)^\circ$ for the ...
kneidell's user avatar
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4 votes
0 answers
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Stability of infinite root systems with a long path in their Coxeter diagrams

Given a Cartan matrix associated to a Coxeter diagram, I can modify it by replacing one of the edges in the diagram with a long chain of vertices connected by simply laced edges; for example, this is ...
Will Dana's user avatar
  • 453
5 votes
1 answer
337 views

Table of products for Lie algebra inner product of roots and weights

For a simple Lie algebra $\frak{g}$, it is usual to scale the inner product so that the shortest simple root has length $2$. With this conventions, where can I find a table (online) of the following ...
Max Schattman's user avatar
5 votes
1 answer
446 views

How to determine a highest weight corresponding to a parabolic subgroup?

Let $G$ be a simply connected, semisimple algebraic group over $\mathbb C$ with maximal torus $T$ and Borel subgroup $B$ containing $T$. If $(V,\pi)$ is an irreducible representation of $G$, then $(V,...
D_S's user avatar
  • 6,100
8 votes
2 answers
569 views

Relationship between $q$-Weyl dimension formula and $q$-analog of weight multiplicity?

$\DeclareMathOperator\dim{dim}$For a dominant (integral) weight $\lambda$ and any (integral) weight $\mu$ of a simple Lie algebra $\mathfrak{g}$, Lusztig's $q$-analog of weight multiplicty $K_{\lambda,...
Sam Hopkins's user avatar
  • 22.7k
9 votes
0 answers
91 views

A characterization of root systems via their intersections with halfspaces

In a recent preprint I obtained a nice characterization of root systems as a side product. I can imagine that this was known before, and that a source for this statement can shorten the proof of my ...
M. Winter's user avatar
  • 12.5k
5 votes
0 answers
209 views

Reference about the root systems $E_{n}$, $n \ge 10$

I am trying to understand the root systems $E_{n}$, $n \ge 10$. In particular, I would like to find some references which describe the number of real roots and imaginary roots of a given degree. ...
Jianrong Li's user avatar
  • 6,101
11 votes
1 answer
309 views

Generalized root systems and reflection groups

Consider the following alternative definition of finite reflection group: Definition: A finite reflection group $\Gamma\subset\mathrm O(\Bbb R^d)$ is a finite group generated by orthogonal ...
M. Winter's user avatar
  • 12.5k
0 votes
0 answers
171 views

What are some nice matrix representations of $E_6$?

I'm planning on doing some SAGE computations to play around with the Lie group $E_6$ (not sure which isogeny class yet), and was wondering if anyone knew of some nice matrix representations of the Lie ...
D_S's user avatar
  • 6,100
4 votes
0 answers
226 views

Do these Zariski-dense subgroups of complex Chevalley group have non-empty intersection with this Bruhat cell?

Let $G$ be a complex Chevalley group (not necessarily adjoint type) with $\operatorname{\mathbb{C}-rank}\geq2$ and let $H$ be a normal subgroup of $G(\mathbb Z)$ with a finite index (so $H$ is Zariski ...
Ami's user avatar
  • 71
4 votes
0 answers
151 views

Is one of the hyperplane partitions of a irreducible root system always generate the whole Weyl group?

Let $\Delta$ be a irreducible root system and $\Delta^+$ be its positive roots. We say a subset $\Delta^{\prime}\subset \Delta^+$ can generate the Weyl group if reflections of roots in $\Delta^{\...
Zhaoting Wei's user avatar
  • 8,657
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
2 votes
1 answer
285 views

The simple reflections of the Weyl group in $\operatorname{SO}_{2n}(\mathbb C)$

Let $W$ be the Weyl group corresponding to the maximal torus $diag(t_1, . . . , t_{n}, t^{−1}_n, . . . , t^{−1}_1)$ in a Borel group of $\operatorname{SO}_{2n}(\mathbb C)$. What are the matrices ...
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
2 votes
1 answer
387 views

Diagonal automorphisms for twisted Chevalley groups

Let $G$ be a Chevalley group over a field $k$ of characteristic $0$. We know that a diagonal automorphism $\phi_h$ of $G$ is of the form $g\mapsto hgh^{-1}$, where $h\in \hat H$ and $\hat H$ ...
Sushil's user avatar
  • 41
2 votes
1 answer
265 views

Root subgroups of simply connected Chevalley groups and their generators

I am looking for a detailed mapping of the root subgroups and elements (and their height) of the simply connected Chevalley groups of type other than $A_n$, and their generators into $\operatorname{GL}...
Ami's user avatar
  • 332
2 votes
1 answer
133 views

Characterization of all $w$ in the Weyl group satisfying $w \geq w_l w_{l, \theta}$

Let $W$ be the Weyl group of a root system $\Phi$ with base $\Delta$ and system of positive roots $\Phi^+$. Let $S = \{ w_{\alpha} : \alpha \in \Delta \}$ be the set of simple reflections ...
D_S's user avatar
  • 6,100
5 votes
1 answer
291 views

Bruhat order and positive roots made negative

Let $(\Phi, V)$ be a reduced root system with base $\Delta$ and Weyl group $W$. Let $\ell$ be the length function of $W$ with respect to the set of simple reflections $S = \{s_{\alpha} : \alpha \in \...
D_S's user avatar
  • 6,100
2 votes
0 answers
106 views

A group generated by all the root subgroups above ideal of $\mathbb Z[\alpha]$ is of finite index in $G(\mathbb Z[\alpha])$

Let $G$ be a simply connected complex Chevalley group and let $T$ be some maximal torus in $G$ with $\dim T\geq 2$. From "A note on generators for arithmetic subgroups of algebraic groups" by ...
Ami's user avatar
  • 332
2 votes
0 answers
64 views

Piecewise linear $\sigma_i$ - notation question

In cluster algebra framework, in order to get root clusters, a modified version of a simple reflection is used. Define $\sigma_i:\Phi_{\geq -1} \to \Phi_{\geq -1}$ by setting: $ \sigma_i(\alpha) = \...
amator2357's user avatar
6 votes
1 answer
415 views

Root lattices and (resolutions of) singular cubic surfaces

(Cross-posted from math.SE since I'm not sure what platform is suitable -- see https://math.stackexchange.com/questions/3331104/root-lattices-and-resolutions-of-singular-cubic-surfaces) Given a ...
modnar's user avatar
  • 501
3 votes
1 answer
321 views

Reflection reverses a root string

I am trying to understand the part of the proof In Fulton's and Harris's Represantation Theory book where he shows that the length of the root string is at most 4: Theorem If $\alpha,\beta$ are roots ...
amator2357's user avatar
4 votes
0 answers
99 views

About the geometry of the set of weights that is strongly linked to $\lambda$

Define $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_\alpha\cdot\lambda<\lambda$ for some $\alpha\in\Phi^+$. More generally, $\eta\uparrow\lambda$ if $\eta=\lambda$ or $\eta=s_{\alpha_1}s_{\...
James Cheung's user avatar
  • 1,855
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
3 votes
2 answers
342 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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