Gelfand–Tsetlin-type weight bases for all special linear Lie algebra representations corresponding to skew Schur functions

Abstract

We generalize the famous weight basis constructions of the finite-dimensional irreducible representations of sl(n,C) obtained by Gelfand and Tsetlin in 1950. Using combinatorial methods, we construct one such basis for each finite-dimensional representation of sl(n,C) associated to a given skew Schur function. Our constructions use diamond-colored distributive lattices of skew-shaped semistandard tableaux that generalize some classical Gelfand–Tsetlin (GT) lattices. Our constructions take place within the context of a certain programmatic study of poset models for semisimple Lie algebra representations and Weyl group symmetric functions undertaken by the first-named author and others. Some key aspects of the methodology of that program are recapitulated here.Combinatorial and representation-theoretic applications of our constructions are pursued here and elsewhere. Here, we extend combinatorial results about classical GT lattices to our more general lattices. We also use classical GT lattices to construct new and combinatorially distinctive weight bases for certain families of irreducible representations of the orthogonal Lie algebras. In another paper, via an entirely similar approach using the non-classical lattices of this paper, we obtain explicit weight bases for some infinite families of irreducible representations of the exceptional simple Lie algebras of types E6 and E7. Other combinatorial applications and generalizations are pursued in companion papers.