An integrality theorem of root systems
Copyright policy )An integrality theorem of root systems
Let \(\Phi\) be an irreducible root system, and \(\Delta\) be a base for \(\Phi\). It is well known that any root in \(\Phi\) is an integral combination of the roots in \(\Delta\). The authors pose a natural question: If \(S\) is a linearly dependent subset of \(\Phi\), can there be a linearly independent subset of \(S\) which generates \(S\)? They prove that any indecomposable subset \(S\) of \(\Phi\) is generated by an indecomposable linearly independent subset of \(S\).
Vector spaces, linear dependence, rank, lineability, Computational Theory and Mathematics, graph theory, root systems, Structure theory for Lie algebras and superalgebras, Geometry and Topology, Simple, semisimple, reductive (super)algebras, indecomposable subset, Trees, Theoretical Computer Science
Vector spaces, linear dependence, rank, lineability, Computational Theory and Mathematics, graph theory, root systems, Structure theory for Lie algebras and superalgebras, Geometry and Topology, Simple, semisimple, reductive (super)algebras, indecomposable subset, Trees, Theoretical Computer Science
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