Weight q-multiplicities for Representations of sp4(C)

Harris, Pamela E.; Lauber, Edward L.; Харрис, Памела Е.; Лаубер, Эдвард Л.

doi:10.17516/1997-1397-2017-10-4-494-502

Cited by 7 publications

(5 citation statements)

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“…Much work has been done in giving closed formulas for Equation (6.4). This includes determining the support of the function when λ is the highest root of a Lie algebra and µ is zero or a positive root [12,21,22,25], determining solutions to associated q-analog problems [23,24,26], and providing visualizations for the support of (6.4) in low rank examples [27,28]. In light of the main results in the current, we pose the following.…”

Section: The Juggling Posetmentioning

confidence: 99%

Kostant's partition function and magic multiplex juggling sequences

Benedetti,

Hanusa,

Harris

et al. 2020

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Kostant's partition function is a vector partition function that counts the number of ways one can express a weight of a Lie algebra g as a nonnegative integral linear combination of the positive roots of 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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Section: The Juggling Posetmentioning

confidence: 99%

Kostant's partition function and magic multiplex juggling sequences

Benedetti,

Hanusa,

Harris

et al. 2020

Preprint

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“…As it turns out, the answer is yes. We prove this next and henceforth we use the substitution for σ = 1 as given in (15).…”

Section: Weyl Alternation Sets Throughout We Letmentioning

confidence: 99%

“…Moreover, for a Lie algebra of rank r, the number of terms appearing in this sum is factorial in the rank. Thus, many have worked to determine closed formulas for the partition function and its q-analog, including low rank examples [7,15,18,19] and for specific families of inputs [9,11,12]. Other work was motivated by the observation that in practice, many terms appearing in computations involving Kostant's weight multiplicity formula are zero [3].…”

Section: Introductionmentioning

confidence: 99%

On Kostant's weight $q$-multiplicity formula for $\mathfrak{sl}_{4}(\mathbb{C})$

Garcia¹,

Harris²,

Loving³

et al. 2020

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The q-analog of Kostant's weight multiplicity formula is an alternating sum over a finite group, known as the Weyl group, whose terms involve the q-analog of Kostant's partition function. This formula, when evaluated at q = 1, gives the multiplicity of a weight in a highest weight representation of a simple Lie algebra. In this paper, we consider the Lie algebra sl4(C) and give closed formulas for the q-analog of Kostant's weight multiplicity. This formula depends on the following two sets of results. First, we present closed formulas for the q-analog of Kostant's partition function by counting restricted colored integer partitions. These formulas, when evaluated at q = 1, recover results of De Loera and Sturmfels. Second, we describe and enumerate the Weyl alternation sets, which consist of the elements of the Weyl group that contribute nontrivially to Kostant's weight multiplicity formula. From this, we introduce Weyl alternation diagrams on the root lattice of sl4(C), which are associated to the Weyl alternation sets. This work answers a question posed in 2019

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“…However, there has been some success in the low rank of some classical Lie algebras (Harris & Lauber, 2017;Refaghat & Shahryari, 2012) and for a particular weight (Harris et al, 2018). Moreover, many methods have been used to solve these types of problems (Adiga et al, 2016;Deckhart, 1985;Harris & Lauber, 2017;Kostant, 1958;Sarikaya et al, 2020;Srivastava & Chaudhary, 2015;Srivastava & Saikia, 2020;Zhang et al, 2009). In this paper, we are interested in finding a closed formula for } in Lie algebras sl 4 ðCÞ and sp 6 ðCÞ.…”

Section: Introductionmentioning

confidence: 99%