2011 / December Volume 6 No.4
Combinatorial Bases For Covariant Representations Of The Lie Superalgebra $gl_{m|n}$
Published Date |
2011 / December
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Title | Combinatorial Bases For Covariant Representations Of The Lie Superalgebra $gl_{m|n}$ |
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Pagination | 415-462 |
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.
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AMS Subject Classification |
17B10
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Received |
2011-06-17
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Accepted |
2011-06-20
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