On connectedness of graphs on direct product of Weyl groups. (English) Zbl 0849.20033

Youssef, Samy A.; Hulsurkar, S. G.

Arch. Math., Brno 31, No. 4, 299-304 (1995).

The graph on Weyl groups has been defined and studied by S. G. Hulsurkar [J. Math. Phys. Sci. 24, No. 6, 363-367 (1990; Zbl 0732.05026)]. The relevant definitions and the results on the Weyl groups can be found in [J. E. Humphreys, Introduction to Lie algebras and representation theory (1972; Zbl 0254.17004)]. In this paper, the connectedness of the graphs on the product of the Weyl groups is investigated. The authors prove that the number of the connected components of the graph on the direct product of the Weyl groups is equal to the product of the numbers of the connected components of the graphs on the factors of the direct product. From this the authors deduce that the graph on the direct product of the Weyl groups is connected iff the graph on each factor of the direct product is connected.

Reviewer: Chen Chengdong (Shanghai)

MSC:

20F55	Reflection and Coxeter groups (group-theoretic aspects)
05C25	Graphs and abstract algebra (groups, rings, fields, etc.)
20E22	Extensions, wreath products, and other compositions of groups

Keywords:

Weyl groups; connectedness; graphs; connected components; direct products

Citations:

Zbl 0732.05026; Zbl 0254.17004