Questions tagged [root-systems]
The root-systems tag has no usage guidance.
257
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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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0
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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 ...
4
votes
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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
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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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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 ...
1
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0
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65
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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 ...
2
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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
$$\left\{
\begin{aligned}
(1+e^{x}(-1+x))^{n-2}&=\...
3
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1
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142
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Action of split torus on positive root spaces
Let $G$ be a connected reductive group over a field $k$ (not necessarily algebraically closed). Let $S$ be a maximal split torus in $G$ with relative root system $\Phi = \Phi_k(S,G)$. Let $\Phi^+$ ...
9
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2
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Which representations of $\mathfrak{sl}(2)$ are homomorphic images of the tensor product of finitely many copies of $\mathbb{C}^2$?
My questions may turn out to be related to Schur functors.
If $\mathfrak{g}$ is a complex semisimple Lie algebra and $\lambda$ is the highest weight of an irreducible representation $V$ of $\mathfrak{...
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Singular del Pezzo surfaces and degeneration of root systems
Let $S$ be a smooth del Pezzo surface of degree $d$ and $K_S^*$ the anticanonical class. It is well known that the set of classes
$$R(S)=\{\alpha\in H^2(S,\mathbb Z)|\alpha^2=-2,\alpha\cdot K_S^*=0\},$...
2
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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 $v\in V\backslash\{0\}$ (...
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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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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, ...
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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 $f\colon\mathbb{P}^1\to X$ ...
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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{...
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1
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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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0
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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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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 ...
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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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0
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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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0
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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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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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1
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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 ...
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2
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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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234
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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 ...
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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 ...
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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 ...
5
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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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1
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446
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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,...
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2
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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,...
9
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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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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.
...
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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 ...
0
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0
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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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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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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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1
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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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1
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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 ...
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1
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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 ...
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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 $\hat H$ ...
2
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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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1
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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 ...
5
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1
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291
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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 \...
2
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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 ...
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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) =
\...
6
votes
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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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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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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_{\...
1
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0
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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,...
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2
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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 $\mathbb{Z}^{\ge 0}$...