Extremal Properties of Bases for Representations of Semisimple Lie Algebras

• Authors: Robert G. Donnelly
• Venue: Journal of Algebraic Combinatorics: An International Journal
• Year: 2003

Let \cal L be a complex semisimple Lie algebra with specified Chevalley generators. Let V be a finite dimensional representation of \cal L with weight basis \cal B. The supporting graph P of \cal B is defined to be the directed graph whose vertices are the elements of \cal B and whose colored edges describe the supports of the actions of the Chevalley generators on V. Four properties of weight bases are introduced in this setting, and several families of representations are shown to have weight bases which have or are conjectured to have each of the four properties. The basis \cal B can be determined to be edge-minimizing (respectively, edge-minimal) by comparing P to the supporting graphs of other weight bases of V. The basis \cal B is solitary if it is the only basis (up to scalar changes) which has P as its supporting graph. The basis \cal B is a modular lattice basis if P is the Hasse diagram of a modular lattice. The Gelfand-Tsetlin bases for the irreducible representations of sl(n, \Bbb C) serve as the prototypes for the weight bases sought in this paper. These bases, as well as weight bases for the fundamental representations of sp(2n, \Bbb C) and the irreducible “one-dimensional weight space” representations of any semisimple Lie algebra, are shown to be solitary and edge-minimal and to have modular lattice supports. Tools developed here are used to construct uniformly the irreducible one-dimensional weight space representations. Similar results for certain irreducible representations of the odd orthogonal Lie algebra o(2n + 1, \Bbb C), the exceptional Lie algebra G2, and for the adjoint and short adjoint representations of the simple Lie algebras are announced.