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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 \(k\).

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
15A04 Linear transformations, semilinear transformations
20F65 Geometric group theory
51N30 Geometry of classical groups
Full Text: DOI
[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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