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Keywords:
root systems; Weyl Groups; connectivity in a graph
root systems; Weyl Groups; connectivity in a graph
Summary:
In this paper, we have studied the connectedness of the graphs on the direct product of the Weyl groups. We have shown 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. In particular, we show 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.
References:
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[3] Verma D.-N.: Role of Affine Weyl Groups in the Representation Theory of Algebraic Chevalley Groups and their Lie Algebras. in “Lie Groups and their Representations", Ed. I.M.Gelfand, Halstead, New York, 1975. MR 0409673 | Zbl 0316.20030
[4] Hulsurkar S.G.: Proof of Verma’s conjecture on Weyl’s dimension polynomial. Inventiones Math., 27(1974), 45. MR 0369555 | Zbl 0298.17005
[5] Narsingh Deo: Graph Theory. Prentice Hall of India, New Delhi, 1990.
[6] Youssef, Samy A., Hulsurkar S.G.: On Connectedness of graphs on Weyl Groups of type $ A_n (n \ge 4) $. Arch. Math. (Brno) 31(1995), 163-170. MR 1368255
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