Combinatorial Bases For Covariant Representations Of The Lie Superalgebra $gl_{m|n}$
by A. I. Molev  

Vol. 6 No. 4 (2011) P.415~P.462

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 $\text{Y}({\mathfrak{gl}}_n)$. One consequence is a new proof of the character formula for the covariant representations first found by Berele and Regev and by Sergeev.

KEYWORDS
Yangian, Gelfand-Tsetlin basis, Mickelsson algebra

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 17B10

MILESTONES

Received: 2011-05-30
Revised : 2011-06-17
Accepted: 2011-06-20

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