Combinatorial bases for covariant representations of the Lie superalgebra gl(m|n)

Details Combinatorial bases for covariant representations of the Lie superalgebra gl(m|n) Combinatorial bases for covariant representations of the Lie superalgebra gl(m|n)

2010-10-04

arxiv.org/abs/1010.0463v2

Bulletin of the Institute of Mathematics, Academia Sinica 6 (2011), 415-462

Mathematics

Representation Theory

40 pages, minor corrections made

Scientific paper

Covariant tensor representations of 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 gl(m|n) in this basis. The basis has the property that the natural Lie subalgebras gl(m) and gl(n) act by the classical Gelfand-Tsetlin formulas. The main role in the construction is played by the fact that the subspace of gl(m)-highest vectors in any finite-dimensional irreducible representation of gl(m|n) carries a structure of an irreducible module over the Yangian Y(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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