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Kostant’s Partition Function and Magic Multiplex Juggling Sequences

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Abstract

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.

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Abbreviations

h :

Height of a juggling state

m :

Hand capacity of a juggling sequence

n :

Length of a juggling sequence

p :

Multiset of positive roots

r :

Rank of Lie algebra

\(s_i\) :

Entry of juggling sequence vector

\(A_r\) :

Lie algebra \(\mathfrak {sl}_{r+1}(\mathbb {C})\)

\(B_r\) :

Lie algebra \(\mathfrak {so}_{2r+1}(\mathbb {C})\)

\(C_r\) :

Lie algebra \(\mathfrak {sp}_{2r}(\mathbb {C})\)

\(D_r\) :

Lie algebra \(\mathfrak {so}_{2r}(\mathbb {C})\)

\(K(\mu )\) :

Number of partitions of \(\mu \) using positive roots of type A

\(K_\Lambda (\mu )\) :

Number of partitions of \(\mu \) using roots from \(\Lambda \)

\(P(\mu )\) :

Set of partitions of \(\mu \)

\(P_\Lambda (\mu )\) :

Set of partitions of \(\mu \) using roots from \(\Lambda \)

\(Q_\Lambda (\mu )\) :

Subset of \(P_\Lambda (\mu )\) subject to a hand capacity constraint

S :

Juggling sequence

\(T_{i,j}\) :

A throw at time i to height j

\(\mathcal {T}\) :

A set of throws

\(\mathcal {T}_r\) :

Set of all throws that land by time \(r+1\)

\(\mathbf {a}\) :

Initial state of a juggling sequence

\(\mathbf {b}\) :

Terminal state of a juggling sequence

\(\mathbf {s}\) :

Juggling state

\(\mathbf {t}\) :

Reflected juggling state

\(\alpha _i\) :

Simple roots of a Lie algebra

\({\tilde{\alpha }}\) :

Highest root of a Lie algebra

\(\beta _i\) :

A positive root

\(\delta (S)\) :

Net change vector of S

\(\varepsilon _i\) :

Standard basis vectors

\(\mu \) :

Weight of a Lie algebra

\(\Gamma \) :

Function between (multisets of) positive roots and (multisets of) throws

\(\Delta \) :

Simple roots of a Lie algebra

\(\Phi \) :

Root system of a Lie algebra

\(\Phi ^+\) :

Positive roots of a Lie algebra

\(\Lambda \) :

Subset of \(\Phi ^+\)

\(\mathrm {JS}\) :

Set of juggling sequences

\(\mathsf {js}\) :

Number of juggling sequences

\(\mathrm {JS}_\mathcal {T}\) :

Set of juggling sequences using throws from \(\mathcal {T}\)

\(\mathsf {js}_\mathcal {T}\) :

Number of juggling sequences using throws from \(\mathcal {T}\)

\(\mathrm {LJS}\) :

Set of labeled juggling sequences

\(\mathsf {ljs}\) :

Number of labeled juggling sequences

\(\mathrm {PJS}\) :

Juggling poset

\(\mathrm {RJS}\) :

Juggling polytope/set of real-valued juggling sequences

\((\cdot ,\cdot ,\cdot )\) :

Parentheses denote a juggling sequence

\(\langle \cdot ,\cdot ,\cdot \rangle \) :

Angle brackets denote a juggling state

\({[}\cdot ,\cdot ,\cdot {]}\) :

Square brackets denote the components of a labeled juggling state

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Acknowledgements

A preliminary version of these results were presented in [48]. This work was supported by the American Institute of Mathematics through their SQuaRE program. We are very appreciative of their support and funding which made this research collaboration possible. We thank Rafael González D’León and Martha Yip for fruitful discussions throughout the process. We also thank Igor Pak and Mark Wilson for their helpful suggestions in Sect. 6.2.2 as well as the anonymous referees. C. Benedetti also thanks grant FAPA of the Faculty of Science at Universidad de los Andes. A. H. Morales received support from NSF Grant DMS-1855536.

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Authors and Affiliations

  1. Departamento de Matemáticas, Universidad de los Andes, Bogotá, Colombia

    Carolina Benedetti

  2. Department of Mathematics, Queens College (CUNY), Flushing, NY, 11367, USA

    Christopher R. H. Hanusa

  3. Department of Mathematics and Statistics, Williams College, Williamstown, MA, 01267, USA

    Pamela E. Harris & Anthony Simpson

  4. Department of Mathematics and Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    Alejandro H. Morales

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  1. Carolina Benedetti
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Correspondence to Pamela E. Harris.

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Benedetti, C., Hanusa, C.R.H., Harris, P.E. et al. Kostant’s Partition Function and Magic Multiplex Juggling Sequences. Ann. Comb. 24, 439–473 (2020). https://doi.org/10.1007/s00026-020-00498-0

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