An integrality theorem of root systems

Let Φ be an irreducible root system and ∆ be a base for Φ; it is well known that any root in Φ is an integral combination of the roots in ∆. In comparison to this fact, we establish the following result: Any indecomposable subset T of Φ is contained in the Z-span of an indecomposable linearly independent subset of T. Throughout this paper, R and Z denote the set of reals and the set of integers respectively and E is a finite dimensional vector space over R with usual innerproduct (∗, ∗). If v0, v1, v2, . . . , vn are vectors in E such that v0 = n i=1 tivi where ti ∈ Z for i = 1, . . . , n, then v0 is called an integral combination of v1, v2, . . . , vn. Let S and T be two subsets of E. If S is contained in the Z-span of T—or equivalently, if every vector of S is an integral combination of vectors in T—then we say that S is generated by T. For occasional graph theoretic terms used in this paper, we refer to [1]. Let S be any subset of E; we can associate a graph with...

• Type: Journal
• Year: 2007
• Where: EJC
• Authors: A. Bhattacharya, G. R. Vijayakumar
• Keywords: EJC 2007, Information Technology, Integral Combination, Irreducible Root, finite Dimensional Vector