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On connectedness of graphs on Weyl groups of type $$A_ n$$ ($$n\geq 4$$). (English) Zbl 0854.20052
The first author [Graphs on Weyl groups, Ph.D. Thesis, Indian Inst. Technol., Kharagpur (1992)] and the authors [Arch. Math. Brno 29, No. 1-2, 19-23 (1993; Zbl 0798.05029)] have defined and studied a graph structure on Weyl groups using the root system associated with them. In the present paper the authors show that such graphs are connected for Weyl groups of type $$A_n$$ ($$n\geq 4$$). The graphs on Weyl groups of type $$A_1$$, $$A_2$$, $$A_3$$ and $$B_2$$ are disconnected. The authors conjecture that the graphs on Weyl groups of irreducible root systems are connected except the four types indicated.
Reviewer: V.L.Popov (Moskva)

##### MSC:
 20F55 Reflection and Coxeter groups (group-theoretic aspects) 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
##### Keywords:
graphs on Weyl groups; irreducible root systems
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