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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 ...
Bobby-John Wilson's user avatar
3 votes
0 answers
166 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 ...
jack's user avatar
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1 vote
1 answer
168 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(...
p6majo's user avatar
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20 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
268 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 ...
Smith's user avatar
  • 73
3 votes
1 answer
100 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 ...
David Schwein's user avatar
1 vote
1 answer
186 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 $\...
Bipolar Minds's user avatar
1 vote
0 answers
66 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 ...
Tim's user avatar
  • 11
6 votes
0 answers
198 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\...
Semen Podkorytov's user avatar
0 votes
0 answers
129 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 ...
user135743's user avatar
1 vote
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113 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 $\eta(s)$ ...
Steven Clark's user avatar
  • 1,091
3 votes
1 answer
160 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) < \...
Andrea B.'s user avatar
  • 355
0 votes
0 answers
43 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-...
Dac0's user avatar
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0 answers
201 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 $V$. ...
Dr. Evil's user avatar
  • 2,711
2 votes
1 answer
235 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 ...
youknowwho's user avatar
2 votes
0 answers
158 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 \...
Eric's user avatar
  • 21
5 votes
0 answers
245 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 ...
Malkoun's user avatar
  • 5,051
0 votes
0 answers
73 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: $$ \begin{aligned} &F(x) = 0\\ &G(x) < 0\\ \end{aligned} $$ ...
AnNam's user avatar
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1 vote
0 answers
43 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 ...
Aleksandr's user avatar
0 votes
1 answer
136 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. ...
Ann's user avatar
  • 43
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 ...
Matt Samuel's user avatar
  • 2,168
3 votes
2 answers
266 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 $W$-...
Henry Talbott's user avatar
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 ...
Lorenzo Del Vecchiopontopolos's user avatar
3 votes
0 answers
121 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 $(\...
Fantas Anadolou's user avatar
7 votes
2 answers
342 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 &...
Марина Marina S's user avatar
7 votes
0 answers
688 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 ...
Malkoun's user avatar
  • 5,051
2 votes
0 answers
144 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 ...
Jake Wetlock's user avatar
  • 1,154
3 votes
2 answers
407 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 ...
Didier de Montblazon's user avatar
3 votes
0 answers
151 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 ...
dm82424's user avatar
  • 360
1 vote
1 answer
133 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 ...
emiliocba's user avatar
  • 2,406
1 vote
1 answer
98 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 ...
Anil Bagchi.'s user avatar
0 votes
1 answer
121 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 ...
bernardorim's user avatar
3 votes
1 answer
154 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 $...
bernardorim's user avatar
2 votes
0 answers
48 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 ...
G. Gallego's user avatar
2 votes
1 answer
182 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 ...
Allen Lee's user avatar
  • 271
2 votes
1 answer
98 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 $\...
freeRmodule's user avatar
  • 1,077
2 votes
1 answer
304 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 $k$-...
Jun Yang's user avatar
  • 381
-1 votes
1 answer
278 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^...
Doaa mahmoud's user avatar
4 votes
0 answers
200 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 ...
Arnold Mckenzie's user avatar
5 votes
1 answer
297 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 ...
Didier de Montblazon's user avatar
2 votes
0 answers
48 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 $\mathfrak{t}$, ...
Malkoun's user avatar
  • 5,051
7 votes
1 answer
220 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 ...
LSpice's user avatar
  • 11.9k
16 votes
2 answers
582 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 ...
LSpice's user avatar
  • 11.9k
2 votes
0 answers
171 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 ...
PSHINH2's user avatar
  • 21
2 votes
0 answers
360 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. ...
Dave Shulman's user avatar
4 votes
0 answers
90 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 $...
Andrew's user avatar
  • 568
5 votes
1 answer
137 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 ...
Alessandro Carotenuto's user avatar
6 votes
1 answer
193 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,...
Mare's user avatar
  • 26.3k
8 votes
0 answers
263 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 ...
Cheng-Chiang Tsai's user avatar
1 vote
1 answer
348 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 ...
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