# Questions tagged [root-systems]

For questions involving abstract root systems, their associated Weyl groups and Dynkin diagrams, as well as their applications to Lie theory, graph theory, or other related fields.

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### Sum of weights of a representation of $U(N)$

Let $R$ be a finite-dimensional irreducible representation of $U(N)$, with the set of weights $W_R$. Each element of $W_R$ is a vector of length $N$ with integer entries. Firstly, I would like to know ...
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### Only Weyl group of rank two root system can be dihedral

Let $\Phi$ be a (reduced, crystallographic) root system, and $W$ its Weyl group. Is it possible to prove that if we know $W$ is dihedral, then the rank of $\Phi$ is two, Without using the ...
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### Simple Reflections on Simple Roots

I have two related questions concerning simple reflections and simple roots. Let $\Phi$ be a root system for a reflection group $W$, let $\Pi \subset \Phi$ be a positive system, and let $\Delta$ be a ...
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### Counting and finding root subsystems

Let $\Phi$ be an irreducible root system. A root subsystem of $\Phi$ is a subset $\Psi \subseteq \Phi$ which is a root system. One can find the possible types of root subsystems of $\Phi$ by deleting ...
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### Type of a root (sub)-system

Let $E:=\{x\in \mathbb{R}^{l+1}:x_1+x_2+\cdots + x_{l+1}=0\}$ and let $\Phi\subseteq E$ denote its root system of type $A_l$ given the basis $\Delta=\{e_i-e_{i+1}, 1\leq i \leq l\}$ and with $\{e_i\}$ ...
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