Questions tagged [root-systems]

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7
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0answers
251 views

Which representations of $\mathfrak{sl}(2)$ are tensor products of powers of irreducible representations?

The so called plethysm problem, from what I understand, deals with decomposing tensor products of representations as direct sums of irreducible representations. I kind of would like to go in the other ...
3
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0answers
110 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\},$...
2
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1answer
112 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 $v\in V\backslash\{0\}$ (...
3
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62 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 ...
3
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1answer
183 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, ...
2
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140 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 $f\colon\mathbb{P}^1\to X$ ...
4
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0answers
95 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{...
1
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1answer
43 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: \...
7
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223 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 ...
9
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0answers
189 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 ...
9
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2answers
275 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, ...
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103 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 ...
0
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103 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 ...
7
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153 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 ...
3
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1answer
129 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 ...
11
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1answer
209 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 ...
4
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1answer
203 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 ...
5
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1answer
151 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 ...
4
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0answers
58 views

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 ...
3
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1answer
160 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 ...
5
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1answer
199 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,...
6
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1answer
306 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,...
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84 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 ...
5
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0answers
154 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. ...
10
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1answer
222 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 ...
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114 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 ...
4
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0answers
206 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 ...
4
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126 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^{\...
4
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1answer
138 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 ...
2
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1answer
161 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 ...
1
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1answer
198 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 ...
2
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1answer
220 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 $\hat H$ ...
1
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1answer
152 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}...
2
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1answer
90 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 ...
4
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1answer
126 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 \...
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0answers
74 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 ...
2
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0answers
54 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) = \...
5
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1answer
239 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 ...
3
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1answer
187 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 ...
4
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93 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_{\...
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0answers
89 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,...
3
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2answers
213 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 $\mathbb{Z}^{\ge 0}$...
2
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0answers
137 views

Intersection of Levi subgroups via intersection of their Weyl groups

Let $G$ be a connected reductive group over $\mathbb{C}$. We fix a maximal torus $T \subset G$. Let $M,L \subset G$ be its Levi subgroups containing $T$ (note that we do note assume that $M,L$ are ...
1
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2answers
114 views

Confusion about $\lambda\in\mathfrak{h}^*$ such that $L(\lambda)\in\mathcal{O}^\mathfrak{p}$

I am reading this paper: Representation type of the blocks of category $\mathcal{O}_S$ On p. 199, it said that While on p. 183 (Section 9.2) of Representations of Semisimple Lie Algebras in the BGG ...
5
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1answer
160 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 ...
6
votes
2answers
224 views

Algorithm to list all Kostant partitions

Let $\Phi_+$ be the set of positive roots in some root system, and let $Q_+$ be the positive part of the root lattice, i.e., the set of elements of the form $\sum_{\beta\in \Phi_+}m_\beta\beta$ with $...
3
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1answer
91 views

Does the Weyl group preserve coprimality in Kac-Moody algebras?

Let $\mathfrak g$ be a Kac-Moody algebra (symmetric, or hyperbolic, or whatever other assumptions you need) with simple roots $\alpha_i$. For $\alpha$ a root, write $\alpha$ in the basis of simple ...
1
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0answers
35 views

Cartan integers for Lie superalgebras

Let $\mathfrak g$ be a basic classical simple Lie superalgebra (BCSLSA in short) with Cartan matrix $A$ of some simple system $\Pi$. Then $A$ satisfies the conditions that 1) $\frac{2a_{ij}}{a_{ii}} \...
2
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0answers
44 views

Denominator identity for Lie superalgebras

Let $\mathfrak g$ be a basic classic simple Lie superalgebra. Fix a maximal isotropic subset $S \subset \Delta$ and choose a set of simple roots $\Pi$ containing $S$. Let $R$ be the Weyl ...
2
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0answers
50 views

Dynkin diagram of Basic classical simple Lie superalgebras

Let $\mathfrak g = \mathfrak g_0 \oplus \mathfrak g_1$ be a basic classical simple Lie superalgebra with the root system $\Delta = \Delta_0 \cup \Delta_1$ and Dynkin diagram $\Gamma$. It is well-known ...