Constructions of Representations of Rank Two Semisimple Lie Algebras with Distributive Lattices

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 conclu...

• Type: Journal
• Year: 2006
• Where: COMBINATORICS
• Authors: L. Wyatt Alverson II, Robert G. Donnelly, Scott J. Lewis, Robert Pervine
• Keywords: COMBINATORICS 2006, Lie Algebra, Semistandard Lattices, Semistandard Posets