Questions tagged [root-systems]
For questions on root systems (the objects classified by Dynkin diagrams).
293 questions
2
votes
0
answers
155
views
Cartan matrix and graph Laplacian
The Cartan matrix for $A_n$ is almost equal (except for the diagonal entry at the endpoints) to the graph Laplacian for its Dynkin diagram. Something similar holds for the other root systems (except ...
- 3,788
0
votes
1
answer
116
views
Find integer coefficients of polynomials from approximate irrational roots [duplicate]
Let take quadratic equations
$$x^2+ax+b=0$$
assume here $a,b$ both are integer and the roots of the equation are irrational if I give you one root in irrational form then is there any method to find $...
- 309
0
votes
0
answers
132
views
Roots and action of the Weyl group
Let $\mathfrak{g}$ be a Kac-Moody algebra a Borel pair $(\mathfrak{b},\mathfrak{t})$, let $R^{+}$ be the set of positive roots, $\alpha_1,\dots,\alpha_n$ the simple roots.
For $\alpha=\sum n_{i}\...
- 3,562
3
votes
4
answers
471
views
Fixed-point subalgebras of automorphisms of $D_4$
Let $\mathfrak{g}$ be a simple complex Lie algebra of type $D_4$, and let $\sigma$ be an automorphism of $\mathfrak{g}$.
Suppose the fixed-point subalgebra $\mathfrak{g}^\sigma$ is simple and of type ...
- 3,003
3
votes
0
answers
183
views
Contracting root data?
I am going to describe a "degeneracy functor"
$$\delta_\mathfrak{q} \ : \ \text{Par}(\text{SL}_n,B_n) \ \to \ \text{Par}(\text{SL}_{n-1},B_{n-1})$$
from parabolics of a big group to a ...
- 6,191
5
votes
0
answers
124
views
Outer automorphisms of simple real Lie algebras
For a simple complex Lie algebra $\mathfrak{g}$, it is well-known that the outer automorphisms of $\mathfrak{g}$ correspond to the automorphisms of its Dynkin diagram. Is a similar result known for a ...
- 377
6
votes
1
answer
256
views
Embeddings of complex simple Lie algebras
We have an embedding of the complex simple Lie algebra $G_2$ into $\mathfrak{so}_7$. Is this embedding unique up to $\mathrm{SO}_7$-conjugacy?
Note it is easy to see the embedding of $G_2$ into $\...
- 3,003
10
votes
3
answers
421
views
A representation of $\frak{g}$ that does not decompose into weight spaces?
Let $\frak{g}$ be a complex semi simple Lie algebra. The category $\mathcal{O}$ consists of special types of $\frak{g}$-representations that have decompositions into weight spaces $M_{\lambda}$, for $\...
5
votes
1
answer
220
views
Constructing a semisimple Lie algebra from a root system without choosing a base
I'm looking for an inverse of the standard map:
$$
(\mathfrak{g}, \mathfrak{h}) \mapsto \Phi
$$
which assigns a root system, to a finite-dimensional semisimple Lie algebra with distinguished Cartan ...
- 1,556
6
votes
2
answers
308
views
Coxeter group action on type C cluster algebra?
A basic question about finite type cluster algebras. Let $A$ denote the cluster algebra of type $\mathrm C_{n-1}$, $n\ge 3$. I will view $A$ as a $\mathbb Z$-algebra with the $n^2$ generators $\Delta_{...
- 3,916
7
votes
0
answers
163
views
Is there a "Weyl Character Formula" for Jacobi Polynomials
Let $R$ be a root system, $R^+$ a choice of positive roots, $W$ the Weyl group, $\Lambda$ its weight lattice, $\Lambda^+$ the cone of dominant weights, and $\rho = \frac{1}{2} \sum_{\alpha \in R^+} \...
- 285
2
votes
0
answers
106
views
Self-contragredient representation for quasi-split Lie group
I am very new to theory of Lie algebras. I have some questions and my study requires those.
Let $\mathrm R$ be a special class of representations such that $\mathrm R$ is absolutely irreducible and $\...
- 185
6
votes
1
answer
280
views
Outer automorphisms of Lie algebras
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $\sigma$ be an automorphism of the Dynkin diagram of $\mathfrak{g}$. If we choose a pinning for $\mathfrak{g}$, we can think of $\sigma$ as an ...
- 3,003
4
votes
1
answer
256
views
Equivalent definitions of the simple $\ell$-roots $A_{i,a}$ of quantum affine algebras
Let $U_q(\widehat{\mathfrak g})$ be the quantum affine algebra over a simple Lie algebra $\mathfrak g$. I am trying to understand and compare the so called simple $\ell$-roots $A_{i,a}$ seen in both ...
- 101
6
votes
0
answers
250
views
Abstract root systems in positive characteristic
Does anyone know of a reference outlining the theory of abstract root systems in positive characteristic?
In the hope of inciting useful remarks, I'll outline how I imagine such a theory might be ...
- 1,556
16
votes
2
answers
642
views
Theta-characteristics on a genus 4 curve and $E_8$
In Symplectization, complexification and mathematical trinities Arnold writes: "I have heard from John McKay that the 27 straight lines on a cubical surface, the 28 tangents of a quartic plane ...
- 39.5k
3
votes
1
answer
246
views
Opposite convex order on the set of positive roots of a semisimple Lie algebra
Let $\mathfrak{g}$ be a semisimple Lie algebra of rank l and let $\Delta^+$ be its set of positive roots. Denote by $s_1,...,s_l$ the simple generators of its Weyl group and let $w_0$ be the longest ...
3
votes
0
answers
255
views
Polynomial from degrees of Weyl group
Let $d_1, \dotsc d_n$ be the degrees (of fundamental invariants) of the Weyl group $W$ of a simple Lie group, (in the reflection representation; see table given on the Wikipedia page for their ...
4
votes
0
answers
155
views
A question about decomposing root system $A_{n}$
Denote $\Phi(n)$ as the root system of Lie algebra $\mathfrak{g}$ of type $A_{n}$. Call a disjoint union $\Phi(n) = \Phi_{1}\sqcup\dotsb\sqcup\Phi_{s}$ a decomposition of $\Phi$ if each $\Phi_{k}$ is ...
- 635
3
votes
1
answer
150
views
Roots of polynomial $\sum_{\sigma \in W} x^{l(\sigma)}$
Let $W$ be Weyl group of a root system $\Phi$ (of finite dimensional simple Lie algebra). For $\sigma\in W$, $l(\sigma)$ be the its length. Consider the following polynomial
$$P_\Phi(x) = \sum_{\sigma ...
- 1,073
4
votes
1
answer
260
views
Compact symmetric spaces and sub-root systems
Given two semisimple complex Lie algebras $\frak{g}$ and $\frak{n}$ such that the root system of $\frak{n}$ arises as a sub-root system of the root system of $\frak{g}$, does this then imply that $\...
4
votes
1
answer
214
views
Relationships between the positive cone inside a root system and the dominant Weyl chamber
Let $G$ be a reductive group and fix a choice of positive roots inside the associated root system.
My question is about the relationship between the cone spanned by $\mathbb{Z}_{\geq 0}$-linear ...
- 43
4
votes
1
answer
115
views
Why does the Athansiadis-Linusson bijection encode floors?
The Athanasiadis-Linusson bijection is a correspondence between dominant regions of the $k$-Shi arrangement (in type A) and $k$-parking functions. I'll take $k=1$ here for convenience here.
Let $V$ be ...
- 535
2
votes
0
answers
111
views
Product of a standard parabolic subgroup with the opposite one
Let $G$ be a semisimple algebraic group defined over an algebraic closed field. Fix a Borel subgroup $B$ and a maximal torus $T\subset B$. Let $\Delta$ be the set of simple roots. For a subset $\...
- 725
3
votes
0
answers
80
views
Root systems of maximally noncomact Cartan subalgebras
Let $G$ be a real reductive Lie group, and $K$ a maximal compact subgroup in $G$. Write $\mathfrak{g}$ for the Lie algebra of $G$, and $\mathfrak{g}=\mathfrak{k}+\mathfrak{p}$ for the Cartan ...
- 961
1
vote
0
answers
335
views
A question about decomposition of irreducible root system
Fix an irreducible root system $\Phi$ with rank $r$ and a root base $\Delta$ (we only care type ADE). Call a disjoint union $\Phi = \Phi_{1}\sqcup\dotsb\sqcup\Phi_{s}$ a decomposition of $\Phi$ if ...
- 635
15
votes
6
answers
869
views
Why, conceptually, does the torus normalizer in $G_2$ split?
Background: Let $G$ be a complex reductive group, $T$ a maximal torus, $N$ the normalizer of $T$ in $G$, and $W = N/T$ the Weyl group. All in all, we have a group extension
$$ 1 \to T \to N \to W \to ...
- 977
3
votes
0
answers
185
views
Does the Bruhat decomposition induces decomposition on integral points (on an open cell)?
Edit: both questions are resolved in comments. Let $F$ be a local field and $O$ its integral points. Let $G$ be a split reductive group over $O$. The Bruhat decomposition states that there is a ...
- 460
4
votes
1
answer
175
views
Sum of two positive roots which is not a root: uniqueness of heights of the summands
Consider a (finite reduced irreducible crystallographic) root system $\Phi$ and four positive roots $\alpha,\beta,\gamma,\delta$ such that $\{\alpha,\beta\} \neq \{\gamma,\delta\}$ and $\alpha+\beta=\...
- 3,530
3
votes
1
answer
160
views
Subgroups of a Weyl group fixing some vectors and its cohomology: MAGMA
I am trying to calculate the number of subgroups of the Weyl group $W(E_N)$ that fix certain vectors $L_i (i = 1,2,3)$ using Magma.
However, the output of the following code (especially #nicesubs) ...
- 1,364
1
vote
1
answer
160
views
Explicit expression of simultaneous resolution of semi-universal deformation of ADE singularity
Denote the semi-universal deformation of ADE singularity by $\mathcal{Y}\to\mathfrak{h}^{\mathbb{C}}/W$, where $\mathfrak{h}^{\mathbb{C}}$ is the complex Cartan algebra of root system of type ADE and $...
- 635
1
vote
0
answers
123
views
The partial orders on the elements of a root system coming from the positive spans of the weights and the roots
Let $(\Delta,V)$ be a root system with a choice of positive roots $\Delta^+$. Denote the $\mathbb{N}_0$-span of the positive roots by $\mathcal{O}^+$, and the $\mathbb{N}_0$-span of the associated ...
3
votes
0
answers
264
views
A property of an irreducible root system
Let $\Phi$ be an irreducible root system. Let $\alpha_k$ be a simple root. I recently observed that the number of positive roots which are bigger than $\alpha_k$ and of height $m$ is same as the ...
- 663
1
vote
1
answer
247
views
Relation between the root lattice of $\mathrm{SO}(7)$ and the root lattice of $G_2$
The root lattice of $\mathfrak{so}(7)$ is given by the following 18 roots:
$$
\left(\begin{array}{c}0\\0\\1\end{array}\right)
,
\left(\begin{array}{c}0\\0\\-1\end{array}\right)
,
\left(...
- 367
21
votes
3
answers
2k
views
Where do root systems arise in mathematics?
One often hears that root systems are ubiquitous in mathematics and physics. The most obvious occurrence of root systems is in the classification of complex simple Lie algebras. Where else do they ...
6
votes
1
answer
332
views
Multiplication factors in folding root systems and Lie algebras by automorphisms
When Stembridge, in the paper Folding by automorphisms, considers folding by automorphism $\sigma$ he considers the root system generated by for each orbit $J$.
$$\sum_{i \in J} \alpha_i .$$
Whereas ...
- 83
4
votes
1
answer
252
views
Are isomorphic maximal tori stably conjugate?
Let $F$ be a field and $G$ a reductive $F$-group. For various applications it is important to understand the "classes" of maximal ($F$-)tori of $G$. Here "class" can mean the ...
- 977
1
vote
1
answer
269
views
Reflections on subspaces of $\text{codim} > 1$
Let $V$ be a real finite-dimensional vector space with inner product $\langle \cdot , \cdot \rangle$.
Let $x,y \in V$ be linearly independent. I was wondering how a reflection $s_{x,y}$ through the $\...
- 1,912
1
vote
0
answers
108
views
Root systems of Weyl groupoids
I am working with the notion of Weyl Groupoids as introduced in "A generalization of Coxeter groups, root systems, and Matsumoto’s theorem" by Heckenberger and Yamane.
The authors generalize ...
- 11
6
votes
0
answers
231
views
Zero-one pairings between sets of vectors
Let
$A\subseteq V$ and
$B\subseteq V^\star$
be spanning sets in
a finite-dimensional real vector space $V$ and
its dual $V^\star$.
Suppose that
$$
\langle b,a\rangle\in\lbrace0,1\rbrace
$$
for all
$a\...
- 819
0
votes
0
answers
187
views
Roots in indefinite lattice of K3 surfaces
Anyone who likes $K3$ surfaces cares about lattices of the form $$ (2d)\cdot y^2 - 2x \cdot z$$ (namely the mukai pairing on $H^*_{alg}(K3)$ of picard $1$ with polarization $d$).
Inside we have ...
- 313
1
vote
0
answers
151
views
Question on recursive formulas for $\eta(2 n+1)$ and $\beta(2 n)$ where $n\in\mathbb{N}$
This question is a refinement of my related MSE question which was asked over 2 years ago and no answers have yet been posted.
Consider the following formulas for the Dirichlet eta function $\eta(s)$ ...
- 1,364
3
votes
1
answer
276
views
Elements of length 0 in extended affine Weyl group for GL(n)
As part of my research, I would like to understand the possible pairs of $(v,\sigma)\in \mathbb Z^n\times S_n$ satisfying the following condition: For $1\le i < j \le n$, we have $\sigma(i) < \...
- 565
0
votes
0
answers
94
views
Relation between real forms of Lie algebras and root systems on pseudoeuclidean vector spaces
This might be trivial but I cannot see it clearly.
Simple complex Lie algebras are fully classified by the root systems arising from the Cartan subalgebra for which the Euclidean norm is the Cartan-...
- 337
0
votes
0
answers
340
views
What does the set of all fundamental coweights look like?
Let $\Phi$ be an irreducible root system in a Euclidean vector space $V$. Let $W$ denote its Weyl group. Choose a base $\Delta=\{\alpha_1,...,\alpha_r\}$ for $\Phi$. Then $\Delta$ is a basis for $V$. ...
- 3,003
2
votes
1
answer
440
views
Stabilizer of a Levi subgroup in the Weyl group and its quotient
(I appologize in advance if this question is too naive for experts, since I know very little about the geometry/combinatorics of Weyl/Coxeter groups.)
For simplicity, let $G$ be a connected reductive ...
- 721
2
votes
0
answers
220
views
Root system terminology
Let $\Phi$ be a root system. In a paper I'm writing, I need to work with subsets $\Phi' \subset \Phi$ satisfying the following two conditions:
For all $\lambda_1,\lambda_2 \in \Phi'$ and $c_1,c_2 \...
- 21
5
votes
0
answers
282
views
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 ...
- 5,377
0
votes
0
answers
124
views
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}
$$
...
- 1
1
vote
0
answers
65
views
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 ...
- 11