Explicit Constructions of the Fundamental Representations of the Symplectic Lie Algebras

Abstract

We give two constructions for each fundamental representation of sp(2n,C). We also present quantum versions of these constructions. These are explicit in the sense of the Gelfand–Tsetlin constructions of the irreducible representations of gl(n,C): we explicitly specify the matrix elements for certain generators of sp(2n,C) with respect to each of the two explicit bases presented. In fact, our constructions appear to have been the first such infinite family of explicit constructions of irreducible representations of simple Lie algebras since the Gelfand–Tsetlin constructions were obtained in 1950. Our approach is combinatorial; the key idea is to find a suitable family of partially ordered sets on which to present the action of the Lie algebra, and then to use these posets to produce the bases and the actions of the generators. Our constructions of the fundamental representations of sp(2n,C) take place on two families of posets which we call “symplectic lattices.” Previously (1999, J. Combin. Theory Ser. A88, 217–234), we used these representation constructions to confirm a conjecture of Reiner and Stanton concerning one of these families of lattices.