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Báo cáo toán học: "Constructions of representations of rank two semisimple Lie algebras with distributive lattices"

Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Constructions of representations of rank two semisimple Lie algebras with distributive lattices. | Constructions of representations of rank two semisimple Lie algebras with distributive lattices L. Wyatt Alverson II leslie.alverson@murraystate.edu Scott J. Lewis scott.lewis@murraystate.edu Robert G. Donnelly rob.donnelly@murraystate.edu Robert Pervine bob.pervine@murraystate.edu Department of Mathematics and Statistics Murray State University Murray KY 42071 USA Submitted Aug 20 2006 Accepted Nov 14 2006 Published Nov 23 2006 Mathematics Subject Classihcation 05E15 Abstract We associate one or two posets which we call semistandard posets to any given irreducible representation of a rank two semisimple Lie algebra over C. Elsewhere we have shown how the distributive lattices of order ideals taken from semistandard posets we call these semistandard lattices can be used to obtain certain information about these irreducible representations. Here we show that some of these semistandard lattices can be used to present explicit actions of Lie algebra generators on weight bases Theorem 5.1 which implies these particular semistandard lattices are supporting graphs. Our descriptions of these actions are explicit in the sense that relative to the bases obtained the entries for the representing matrices of certain Lie algebra generators are rational coefficients we assign in pairs to the lattice edges. In Theorem 4.4 we show that if such coefficients can be assigned to the edges then the assignment is unique up to products we conclude that the associated weight bases enjoy certain uniqueness and extremal properties the solitary and edge-minimal properties respectively . Our proof of this result is uniform and combinatorial in that it depends only on certain properties possessed by all semistandard posets. For certain families of semistandard lattices some of these results were obtained in previous papers in Proposition 5.6 we explicitly construct new weight bases for a certain family of rank two symplectic representations. These results are used to help obtain in Theorem .