Extremal properties of bases for representations of semisimple Lie algebras

Donnelly, Robert G.

Journal of Algebraic Combinatorics, Tome 17 (2003) no. 3, pp. 255-282.

Résumé

Summary: Let $Lscr$ be a complex semisimple Lie algebra with specified Chevalley generators. Let $V$ be a finite dimensional representation of $Lscr$ with weight basis $bernou$. The supporting graph P of $bernou$ is defined to be the directed graph whose vertices are the elements of $bernou$ 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 $bernou$ 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 $bernou$ is $solitary$ if it is the only basis (up to scalar changes) which has $P$ as its supporting graph. The basis $bernou$ is a modular lattice basis if $P$ is the Hasse diagram of a modular lattice. The Gelfand-Tsetlin bases for the irreducible representations of $sl( n, Copf)$ serve as the prototypes for the weight bases sought in this paper. These bases, as well as weight bases for the fundamental representations of $s$p($2 n, Copf$) and the irreducible $ldquo$one-dimensional weight space $rdquo$ 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 $o(2 n + 1, Copf)$, the exceptional Lie algebra $G _{2}$, and for the adjoint and short adjoint representations of the simple Lie algebras are announced.

Détail

Keywords: semisimple Lie algebras, irreducible representations, supporting graphs