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On the adjoint representation of and the Fibonacci numbers. (English. French summary) Zbl 1273.17010
Summary: We decompose the adjoint representation of \(\mathfrak{sl}_{r+1} = \mathfrak{sl}_{r+1}(\mathbb C)\) by a purely combinatorial approach based on the introduction of a certain subset of the Weyl group called the Weyl alternation set associated to a pair of dominant integral weights. The cardinality of the Weyl alternation set associated to the highest root and zero weight of \(\mathfrak{sl}_{r+1}\) is given by the \(r\)th Fibonacci number. We then obtain the exponents of \(\mathfrak{sl}_{r+1}\) from this point of view.

17B10 Representations of Lie algebras and Lie superalgebras, algebraic theory (weights)
11B39 Fibonacci and Lucas numbers and polynomials and generalizations
Full Text: DOI
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