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.