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Reflection processes on graphs and Weyl groups. (English) Zbl 0741.05035
Let us consider a weighted graph with the set of vertices \(V=\{0,1,\dots,n-1\}\) and the set of edges \(E\), \(\phi(i,j)\) being the weight of the edge \((i,j)\) (a nonnegative integer). For a function \(f: V\to\mathbb{R}\), assuming that \(f(i)<0\), the transformation \(T_ i: \mathbb{R}^ n\to\mathbb{R}^ n\) is defined as follows: \[ T_ i(f)(j)=\begin{cases} -f(i) &\text{for \(j=i\)}\\ f(j)+\phi(i,j)f(i) &\text{for \((i,j)\in E\)}\\ f(j) &\text{for \((i,j)\not\in E, i\neq j\).}\end{cases} \] Conditions are found under which, after using these transformations several times, one obtains such a function \(g\) with \(g(k)\geq 0\) for all \(k\).

MSC:
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
15A04 Linear transformations, semilinear transformations
20F65 Geometric group theory
51N30 Geometry of classical groups
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References:
[1] Kac, V. G., Infinite Dimensional Lie Algebras (1983), Birkhäuser: Birkhäuser Basel · Zbl 0574.17002
[2] Bourbaki, N., Groupes et algèbres de Lie (1968), Herman: Herman Paris, ch. 4-6
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