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Diamond representations of \({\mathfrak {sl}}(n)\). (English) Zbl 1120.17005

In a previous paper, one of the authors, N. J. Wildberger, had introduced a new model for the irreducible representations of \(\mathfrak{s}\mathfrak{l}(3)\). To this end, he had defined certain polytopes in three dimensional space, called diamonds, and used them to construct such representations. In the paper under review, the authors extend this method to get representations of \(\mathfrak{s}\mathfrak{l}(n)\) and show that the diamond cone module is a quotient of the shape algebra. They provide a number of examples and explicit computations.

MSC:

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
05E10 Combinatorial aspects of representation theory
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References:

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