## 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.

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 Friday, October 8, 2010