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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
##### Keywords:
reflection processes; Weyl groups; transformations
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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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