# Questions tagged [root-systems]

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186
questions

**7**

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122 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**

votes

**1**answer

106 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 ...

**9**

votes

**1**answer

155 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**

votes

**1**answer

167 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 ...

**4**

votes

**1**answer

105 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**

votes

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51 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**

votes

**1**answer

118 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 ...

**4**

votes

**1**answer

131 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**

votes

**1**answer

237 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,...

**9**

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79 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**

votes

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139 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.
...

**9**

votes

**1**answer

191 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 ...

**0**

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96 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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204 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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89 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**

votes

**1**answer

120 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**

votes

**1**answer

153 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**

vote

**1**answer

186 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 ...

**1**

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**1**answer

180 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**

vote

**1**answer

88 views

### Root subgroups of simply connected Chevalley groups and their generators

I'm looking for a detailed mapping of the root subgroups and elements(and their hight) of the simply connected Chevalley groups of type other than $A_n$, and their generators into $\operatorname{GL}_n(...

**2**

votes

**1**answer

82 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**

votes

**1**answer

98 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 \...

**1**

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68 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 ...

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53 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) =
\...

**4**

votes

**1**answer

172 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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**1**answer

152 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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91 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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83 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**

votes

**2**answers

184 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 ...

**2**

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107 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 ...

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vote

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107 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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**1**answer

149 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 ...

**3**

votes

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181 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**

votes

**1**answer

84 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 ...

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33 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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41 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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40 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 ...

**4**

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93 views

### Panyushev's conjectured duality for root poset antichains

In his 2004 paper "ad-nilpotent ideals of a Borel subalgebra: generators and duality" (https://www.sciencedirect.com/science/article/pii/S0021869303006380), Panyushev conjectured (Conjecture 6.1) the ...

**5**

votes

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328 views

### For a fixed dominant weight $\lambda$, are almost all dominant weights in the same coset above it?

First some notation as in e.g. the book by Humphreys on Lie Algebras.
Let $E$ be an Euclidean space with inner product $(-,-)$, and denote $\langle v,w \rangle = \frac{2(v,w)}{(w,w)}$. Let $\Phi$ be ...

**0**

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**1**answer

68 views

### In Type $A$, if the Bruhat graph of an element $w$ in the Weyl group is regular, then to show that $l(w)=$ # $ \{\alpha \in R^+| s_{\alpha} \le w\}$

I am trying to prove that for type $A$ , rational smoothness of Schubert varieties implies smoothness.
So suppose we are in Type $A_{n-1}$, so let $G=Sl(n,\mathbb C)$, $B=$ the group of upper ...

**0**

votes

**1**answer

144 views

### Weyl Group Element $w$ fixing a root, and its presentation as product of simple reflections $w=s_1\dots s_n$

Let $\Phi$ be a root system and $\gamma \in \Phi$ a root. Let $W$ be the Weyl group and $\Delta$ a set of simple roots. Let $w \in W$ such that $w(\gamma)=\gamma$. Is it true that if $w=s_1\dots s_n$ ...

**5**

votes

**3**answers

448 views

### Existence of a weight of a representation in the fundamental Weyl chamber

Let $\mathfrak g$ be a complex simple Lie algebra.
Fix a Cartan subalgebra $\mathfrak h$ of $\mathfrak g$, let $\Delta$ denote the corresponding root system.
Pick a partial order on $\mathfrak h$, ...

**4**

votes

**2**answers

454 views

### Non-faithful irreducible representations of simple Lie groups

For a complex simple Lie algebra $\frak{g}$, which of its finite dimensional irreducible representations give non-faithful representations of the corresponding simply-connected compact Lie group.
...

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90 views

### $q$-Kostant partition function and flow polytopes?

The Kostant partition function is known to be related to volumes and Ehrhart polynomials of flow polytopes of graphs (see e.g. https://link.springer.com/article/10.1007/s00031-008-9019-8 or https://...

**8**

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258 views

### Gram matrix determinant in dimension 4 and $E_8$

Consider a determinant of a Gram matrix in dimension $4$.
$$\begin{vmatrix}
1 & -\cos(\alpha_1) & -\cos(\alpha_2) & -\cos(\alpha_3)\\
-\cos(\alpha_1) & 1 & -\cos(\alpha_6)& -\...

**3**

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72 views

### Description of real roots of Kac—Moody algebra

Let $\Delta$ be a root system associated to a generalized Cartan matrix, $\alpha_1,\ldots,\alpha_n$ its simple roots.
It is known that if $\Delta$ is of finite, affine or hyperbolic type, $\alpha=\...

**5**

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180 views

### Can the Weyl orbits of fundamental weights tell us the Cartan matrix?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra, $\Delta$ its root system contained in $\mathfrak{t}^{\vee}$ for a Cartan sub-algebra $\mathfrak{t}$ of $\mathfrak{g}$. Let $W$ be its Weyl group....

**1**

vote

**1**answer

148 views

### Duality isomorphism of representations of the maximal torus with respect to Steinberg's basis—is it an involution?

I am trying to apply Steinberg's basis of his paper "On a theorem of Pittie" (MSN) for the case $G$ of type $A_2$ and the maximal torus $T$ itself as a maximal rank subgroup. Denote by $\alpha_1, \...

**0**

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71 views

### Associated subgroup of Weyl group

Let $\Phi$ be a root system.
For a weight $\lambda\in\mathfrak{h}^*$,
start by defining
$
\Phi_{[\lambda]}:=\{\alpha\in \Phi \ | \ \langle \lambda,\alpha^{\lor}\rangle\in\mathbb{Z} \}
$
and
$
W_{[\...

**3**

votes

**1**answer

58 views

### Reference request: existence of a subgroup of $G(\mathcal O_k)$ that is “uniform” across $P \overline{N}$

Let $G$ be a connected, reductive group over a $p$-adic field $k$. Let $P_0$ be a minimal parabolic subgroup of $G$ containing a maximal split torus $A_0$. Let $K$ be a maximal compact open subgroup ...