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Kostant’s partition function and magic multiplex juggling sequences. (English) Zbl 1451.05243
Summary: Kostant’s partition function is a vector partition function that counts the number of ways one can express a weight of a Lie algebra \(\mathfrak{g}\) as a nonnegative integral linear combination of the positive roots of \(\mathfrak{g} \). Multiplex juggling sequences are generalizations of juggling sequences that specify an initial and terminal configuration of balls and allow for multiple balls at any particular discrete height. Magic multiplex juggling sequences generalize further to include magic balls, which cancel with standard balls when they meet at the same height. In this paper, we establish a combinatorial equivalence between positive roots of a Lie algebra and throws during a juggling sequence. This provides a juggling framework to calculate Kostant’s partition functions, and a partition function framework to compute the number of juggling sequences. From this equivalence we provide a broad range of consequences and applications connecting this work to polytopes, posets, positroids, and weight multiplicities.
MSC:
05E10 Combinatorial aspects of representation theory
05A15 Exact enumeration problems, generating functions
05A18 Partitions of sets
17B22 Root systems
17B99 Lie algebras and Lie superalgebras
Software:
lie_algebras
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