Questions tagged [root-systems]
The root-systems tag has no usage guidance.
256
questions
0
votes
1
answer
162
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 $\...
0
votes
0
answers
49
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 ...
6
votes
0
answers
185
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\...
0
votes
0
answers
115
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 ...
1
vote
0
answers
106
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 $\eta(s)$ ...
3
votes
1
answer
138
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) < \...
0
votes
0
answers
39
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-...
0
votes
0
answers
166
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 $V$. ...
2
votes
1
answer
191
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 ...
2
votes
0
answers
155
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 \...
5
votes
0
answers
243
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 ...
0
votes
0
answers
70
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
vote
0
answers
42
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 ...
0
votes
1
answer
119
views
Calculating relative root systems
Let $\mathbf{G}$ be a connected semisimple algebraic group defined over a field $k$. Let $T$ be a maximal torus of $\mathbf{G}$ defined over $k$, and let $S \subset T$ be a maximal $k$-split torus. ...
1
vote
1
answer
71
views
Linear independence of reciprocals of products of closed sets of roots in type $A$ inversion sets
Consider the root system $R$ for a Coxeter system $(W,S)$ of type $A_n$ with a choice of simple roots. Denote by $I(w)$ for $w\in W$ the set of positive roots $\beta\in R^+$ such that $w(\beta)$ is a ...
3
votes
2
answers
206
views
Describing characters of a reductive group in terms of characters of a maximal torus
Say I have a reductive complex algebraic group $G$ with maximal torus $T$ and associated Weyl group $W$. I would like to be able to say that the characters of $G$ are in bijection with the $W$-...
1
vote
0
answers
105
views
Weyl group action on the Lie algebra [duplicate]
Let $W$ be the Weyl group of a complex semisimple Lie algebra $\frak{g}$. Certainly $W$ acts on the root system $R$ of $\frak{g}$ but can it be made to act on $\frak{g}$ or on the universal enveloping ...
3
votes
0
answers
112
views
Root space inner products and the partial order on roots
For a root system $R$ and a choice of positive roots $R^+$ it is a standard fact (see, e.g., Bourbaki, "Lie Groups and Lie Algebras," Theorem 1 of Section 1.3 of Chapter VI) that
if $(\...
7
votes
1
answer
331
views
Relation between different $E_8$ matrices
There are several rank-8 square matrices known to be related to $E_8$:
Cartan $E_8$ matrix https://en.wikipedia.org/wiki/E8_(mathematics)#Cartan_matrix
$$M_1=\left [\begin{array}{rr}
2 & -1 &...
7
votes
0
answers
675
views
Is this construction related to the geometric Langlands program perhaps?
Given a complex Lie algebra $\mathfrak{g}$, a choice of Cartan subalgebra $\mathfrak{h}$ of $\mathfrak{g}$ and a dominant integral weight $\lambda$ of $\mathfrak{g}$, there is a natural construction ...
2
votes
0
answers
133
views
The Cartan is a complex vector space but the root system is real?
Let $\frak{g}$ be a complex semisimple Lie algebra with some choice of Cartan subalgebra $\frak{h}$. The dual space $\frak{h}^* = \mathrm{Hom}_{\mathbb{C}}(\frak{h},\mathbb{C})$ is a complex vector ...
3
votes
2
answers
348
views
Pairing a root with the half-sum of positive roots
Let $\frak{g}$ be a finite-dimensional complex simple Lie algebra together with a choice of Cartan subalgebra and associated root system $(\Delta, (-,-))$. Also we denote the half-sum of positive ...
3
votes
0
answers
149
views
Disconnected reductive algebraic groups in Sage
All simply connected split simple groups have been implement on Sage and it is possible to find their highest roots, fundamental weights, Dynkin diagrams or compute the tensor of two of their ...
1
vote
1
answer
126
views
About certain elements in the zero weight space of an irreducible representation of the complex simple Lie algebra of type G$_2$
$\newcommand{\fg}{\mathfrak g}\newcommand{\ee}{\varepsilon}$Let $\fg$ be the complex simple Lie algebra of type G$_2$.
We consider its root system as follows (though it is probably not necessary to ...
1
vote
1
answer
97
views
Problem in understanding a fact about Belavin-Drinfeld triple
A Belavin-Drinfeld triple associated to a simple Lie algebra $L$ is a triple $(\Gamma_1, \Gamma_2, \tau)$ where $\Gamma_1, \Gamma_2 \subseteq \Gamma$ ($\Gamma$ is a set of simple roots or fundamental ...
0
votes
1
answer
117
views
Sub-coroot lattices
[This is a sequel to the previous question sub-coroot systems, that has been answered! :-) ]
Let $T$ be a maximal torus of a compact Lie group $K$,
and let $\Lambda \subset {\mathfrak t}$ be the ...
3
votes
1
answer
149
views
Sub-coroot systems
Let $T$ be a maximal torus of a compact Lie group $K$,
and let $\Psi \subset {\mathfrak t}$ be the (finite) set of coroots for $(K,T)$, where $\mathfrak t$ is the Lie algebra of $T$.
Assume now that $...
2
votes
0
answers
47
views
Multiplicative invariants of non-reduced root systems
It is a well known fact (cf. [1] VI.3.4 Thm. 1) that if $\Phi$ is a (reduced) root system with weight lattice $P$ and $W$ is the Weyl group of this root system, then the algebra of invariant ...
2
votes
1
answer
179
views
Find an analogue of Weyl chamber structure
Let $G$ be a split reductive group and let $T$ be a split maximal torus whose rank is $l$. Is it possible to find a base $\gamma_1,..., \gamma_l$ of the weight lattice $X(T)$ such that the cone $C$ in ...
2
votes
1
answer
88
views
Real roots along root strings
Let $A$ be a Cartan matrix, i.e. a $n\times n$ matrix with integer entries such that $A_{ii}=2$ and $A_{ij}\leq0$ for $i\neq j$. Then the corresponding Kac-Moody Lie algebra has a Cartan subalgebra $\...
2
votes
1
answer
275
views
Tensor product of fundamental representations
Let $\mathfrak{g}$ be a simple complex Lie algebra. Let $V_1,\cdots, V_n$ be the fundamental representations (the irreducible ones with fundamental weights $\omega_1,\cdots,\omega_n$). Take a $k$-...
-1
votes
1
answer
214
views
finding positive roots for a polynomial [closed]
I have a polynomial, and I want to get the conditions for the number of positive roots
What are the different methods out there to determine these conditions?
this is the polynomial:
f(g)=A1g^5 + A2g^...
4
votes
0
answers
190
views
Schur polynomials are polynomials but also sequences on a lattice?
Monomial symmetric polynomials in $n$ variables $x_1, \ldots x_n$ form a natural basis for the space $\mathcal{S}_n$ of symmetric polynomials in $n$ variables and are defined by additive ...
5
votes
1
answer
277
views
Non-standard partial orders on root systems
Let $\frak{g}$ be a semisimple complex Lie algebra and let $\Delta$ be its associated root system with $\{\alpha_1, \dotsc, \alpha_l\}$ a choice of positive roots. As we all know - $\Delta$ admits a ...
2
votes
0
answers
47
views
A construction of Weyl-equivariant maps from the space of regular Cartan triples to the space of tuples of complex polynomials (up to scalar factors)
Let $G$ be a compact semisimple Lie group and let $T$ be a maximal torus in $G$. On the Lie algebra level, we have a real Lie algebra $\mathfrak{g}$ and a (particular) real slice, say $\mathfrak{t}$, ...
7
votes
1
answer
215
views
Why is the fundamental group of $\mathsf E_n$ cyclic of order $9 - n$?
Several years ago, I mentioned offhandedly to a colleague that I had noticed that, if you extend the $\mathsf E_n$ series downwards in the natural way, by removing nodes from the long arm of $\mathsf ...
16
votes
2
answers
540
views
Typos in Bourbaki's root-system tables
A while ago, a colleague told me that he thought he remembered that there were typos in Bourbaki's tables in the English translation of "Groupes et algèbres de Lie", but that he could no ...
2
votes
0
answers
159
views
Root systems and subroot systems
Given the root system $E_{6}$ with basis $\alpha_{1},\dotsc,\alpha_{6}$. How would I find all subroot systems of $E_{6}$ (up to Weyl equivalence) where I can write the basis of each subroot system in ...
2
votes
0
answers
323
views
The Weyl dimension formula for fundamental weights
The Weyl dimension formula is an equation to calculate the dimension of a simple $\frak{g}$-module $V_{\lambda}$, of highest weight $\lambda$, for $\frak{g}$ a complex semisimple Lie algebra. ...
4
votes
0
answers
89
views
Interpretation of the coefficients in the sum of positive roots
Take a finite Cartan datum with index set $I$, simple roots $\{\alpha_i\mid i \in I\}$ and positive roots $\Phi^+$. Let $2\rho=\sum_{\alpha\in\Phi^+}\alpha$ be the sum of the positive roots and write $...
5
votes
1
answer
130
views
PBW basis for the quantized enveloping Lie algebra of $\mathfrak{g}_2$
I would like to know if you have any reference where I can find the canonical PBW basis for $U_q(\mathfrak{g}_2),$ computed using the action of the braid group as defined by Luzstig.
Alternatively I ...
6
votes
1
answer
189
views
Order ideals of positive root systems and avoiding group elements in the Weyl group
Let $X$ be the poset of positive roots of a finite root system of Dynkin type $Q$.
Question 1: In Dynkin type $A_n$, is it true that the poset of order ideals of $X$ is isomorphic to the poset of [2,...
8
votes
0
answers
257
views
A Lie-theoretic question regarding $B\ltimes \mathfrak{g}/\mathfrak{b}$
I am stuck on a seeming elementary Lie-theoretic question arising from a study of components of affine Springer fibers. Will be very grateful if somebody would like to share some insight, or ...
1
vote
1
answer
334
views
The principal congruence subgroup of the symplectic group over the integers
Consider the symplectic group $\text{Sp}_{2g}(\mathbb{Z})$ over the integers. It has a classical root system $C_g$ and associated root subgroups $U_\varphi$ for $\varphi\in C_g$. These subgroups are ...
3
votes
0
answers
89
views
How to determine sublattices S of a root lattice R
Let $R$ be a root lattice of a irreducible root system $\Phi$.
Suppose $W$ is a Weyl group of $\Phi$ and $S$ is a sublattice of $R$ which is $W$-stable and satisfies $|R/S|<\infty$.
For example, ...
1
vote
1
answer
255
views
A nice/simple relationship between the Chevalley generators of $\mathfrak{sp}_n$ and the Chevally generators of $\mathfrak{sl}_n$?
The Lie algebra $\mathfrak{sl}_n$ is defined to be the trace free matrices in $M_n(\mathbb{C})$. The Lie algebra $\mathfrak{so}_n$ is defined to be the matrices $A$ in $M_n(\mathbb{R})$ satisfying $A +...
4
votes
1
answer
175
views
Structure of the permutation groups acting on the root systems of Niemeier lattices of type $A_{k}^n$
I have been doing research on the Niemeier lattices with root systems of type, $A_{k}^n$ and I am particularly interested in the finite groups permuting the constituent root systems. These groups ...
1
vote
0
answers
227
views
Condition for a sum of images of fundamental dominant weights to lie on a wall
Let $\Delta$ be a system of simple roots in a root system with Weyl group $W$. For $\alpha\in\Delta$, let $\varpi_\alpha$ be the corresponding fundamental dominant weight. Let $w\neq r$ be elements of ...
4
votes
0
answers
96
views
Why is the $A$-series root system best written in a vector space of one dimension higher?
In the classification of root systems, we have four families $A_n,B_n,C_n$, and $D_n$, and six exceptionals $E_6,E_7,E_8, F_4$, and $G_2$. For every non-exceptional case except $A_n$, the root system ...
4
votes
1
answer
390
views
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 ...