Questions tagged [root-systems]
The root-systems tag has no usage guidance.
263
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A weakening of the definition of positive roots for a root system
Given a root system $\Delta$ a choice of positive/negative roots is a decomposition of the elements of $\Delta$ into two subsets $\Delta^+$ and $\Delta^-$ satisfying
$$\Delta^+ = - \Delta^-\tag{$*$}\...
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0
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58
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The partial orders on the elements of a root system coming from the positive spans of the weights and the roots
Let $(\Delta,V)$ be a root system with a choice of positive roots $\Delta^+$. Denote the $\mathbb{N}_0$-span of the positive roots by $\mathcal{O}^+$, and the $\mathbb{N}_0$-span of the associated ...
3
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169
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A property of an irreducible root system
Let $\Phi$ be an irreducible root system. Let $\alpha_k$ be a simple root. I recently observed that the number of positive roots which are bigger than $\alpha_k$ and of height $m$ is same as the ...
- 611
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1
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Relation between the root lattice of $\mathrm{SO}(7)$ and the root lattice of $G_2$
The root lattice of $\mathfrak{so}(7)$ is given by the following 18 roots:
$$
\left(\begin{array}{c}0\\0\\1\end{array}\right)
,
\left(\begin{array}{c}0\\0\\-1\end{array}\right)
,
\left(...
- 355
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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 ...
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1
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268
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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 ...
- 73
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101
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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 ...
- 509
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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 $\...
- 1,743
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66
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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
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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\...
- 799
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129
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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 ...
- 293
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113
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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 $\eta(s)$ ...
- 1,091
3
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1
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160
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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) < \...
- 355
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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-...
- 295
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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 $V$. ...
- 2,711