Combinatorial bases for covariant representations of the Lie superalgebra \( \mathfrak{gl}_{m|n} \)

A. I. Molev

Abstract

Covariant tensor representations of \( \mathfrak{gl}_{m|n} \) occur as irreducible components of tensor powers of the natural \( (m+n) \)-dimensional representation. We construct a basis of each covariant representation and give explicit formulas for the action of the generators of \( \mathfrak{gl}_{m|n} \) in this basis. The basis has the property that the natural Lie subalgebras \( \mathfrak{gl}_m \) and \( \mathfrak{gl}_n \) act by the classical Gelfand-Tsetlin formulas. The main role in the construction is played by the fact that the subspace of \( \mathfrak{gl}_m \)-highest vectors in any finite-dimensional irreducible representation of \( \mathfrak{gl}_{m|n} \) carries a structure of an irreducible module over the Yangian \( Y(\mathfrak{gl}_n) \). One consequence is a new proof of the character formula for the covariant representations first found by Berele and Regev and by Sergeev.