# zbMATH — the first resource for mathematics

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
Full Text:
##### References:
  Kac, V. G., Infinite Dimensional Lie Algebras (1983), Birkhäuser: Birkhäuser Basel · Zbl 0574.17002  Bourbaki, N., Groupes et algèbres de Lie (1968), Herman: Herman Paris, ch. 4-6
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.