Kostant’s weight multiplicity formula and the Fibonacci and Lucas numbers

Chang, Kevin L.; Harris, Pamela E.; Insko, Erik

doi:10.4310/joc.2020.v11.n1.a7

Cited by 8 publications

(11 citation statements)

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“…Note that specializing Theorem 1.2 to = 1 and µ = 0 recovers the result presented in [3,Theorem 3.1], which shows that the cardinality of the associated Weyl alternation set is given by a Fibonacci number. Extending our results to the remaining pairs of weights listed in List A and List B is much more difficult since the Weyl group action on the highest weight of these representations is not as straightforward to describe.…”

Section: Introductionsupporting

confidence: 71%

“…In this case, Harris, Insko, and Williams established that the cardinalities of the Weyl alternation sets A(λ, 0) are given by linear recurrences with constant coefficients [7,8,12]. In 2017, Chang, Harris, and Insko described the sets A(β, µ), where β is the sum of all of the simple roots in a classical Lie algebra and µ is an integral weight, and showed that the cardinality of these sets are given by the Fibonacci numbers or multiples of the Lucas numbers [3].…”

Section: Introductionmentioning

confidence: 99%

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When is the $q$-Multiplicity of a Weight a Power of $q$?

Harris¹,

Rahmoeller

Schneider

et al. 2019

Electron. J. Combin.

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Berenshtein and Zelevinskii provided an exhaustive list of pairs of weights (λ, µ) of simple Lie algebras g (up to Dynkin diagram isomorphism) for which the multiplicity of the weight µ in the representation of g with highest weight λ is equal to one. Using Kostant's weight multiplicity formula we describe and enumerate the contributing terms to the multiplicity for subsets of these pairs of weights and show that, in these cases, the cardinality of these contributing sets is enumerated by (multiples of) Fibonacci numbers. We conclude by using these results to compute the associated q-multiplicity for the pairs of weights considered, and conjecture that in all cases the q-multiplicity of such pairs of weights is given by a power of q.

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Section: Introductionsupporting

confidence: 71%

Section: Introductionmentioning

confidence: 99%

When is the $q$-Multiplicity of a Weight a Power of $q$?

Harris¹,

Rahmoeller

Schneider

et al. 2019

Electron. J. Combin.

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“…In [2] #getting coefficients bigToSmall = { "A" : {"a", "d", "j"}, "B" : {"b", "d", "j"}, "C" : {"a", "e", "j"}, "D" : {"a", "d", "l"}, "E" : {"c", "e", "j"}, "F" : {"b", "f", "j"}, "G" : {"a", "g", "l"}, "H" : {"b", "d", "l"}, "I" : {"a", "e", "o"}, "J" : {"c", "f", "j"}, "K" : {"b", "h", "l"}, "L" : {"a", "i", "o"}, "M" : {"b", "f", "p"}, "N" : {"c", "e", "o"}, "O" : {"a", "g", "q"}, "P" : {"c", "f", "p"}, "Q" : {"a", "i", "q"}, } #contradictions of type III zeroToOne = { "b":{"a"}, "c":{"a", "b", "f"}, "e":{"d"}, "f":{"d", "e"}, "g":{"d","e"}, "h":{"d","e","f","g"}, "i":{"d","e","g"}, "l":{"j"}, "o":{"j","l"}, "p":{"j","l","o"}, "q":{"j","l","o","i"}, "a":set(), "d": set(), "j": set(), "p": set(), A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q "A" : {"a", "d", "j"}, "B" : {"b", "d", "j"}, "C" : {"a", "e", "j"}, "D" : {"a", "d", "l"}, "E" : {"c", "e", "j"}, "F" : {"b", "f", "j"}, "G" : {"a", "g", "l"}, "H" : {"b", "d", "l"}, "I" : {"a", "e", "o"}, "J" : {"c", "f", "j"}, "K" : {"b", "h", "l"}, "L" : {"a", "i", "o"}, "M" : {"b", "f", "p"}, "N" : {"c", "e", "o"}, "O" : {"a", "g", "q"}, "P" : {"c", "f", "p"}, "Q" : {"a", "i", "q"} {"a"}, "c":{"a", "b", "f"}, "e":{"d"}, "f":{"d", "e"}, "g":{"d","e"}, "h":{"d","e","f","g"}, "i":{"d","e","g"}, "l":{"j"}, "o":{"j","l"}, "p":{"j","l","o"}, "q":{"j","l","o","i"}, "a":set(), "d": set(), "j": set(), "p": set(), A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q "A" : {"a", "d", "j"}, "B" : {"b", "d", "j"}, "C" : {"a", "e", "j"}, "D" : {"a", "d", "l"}, "E" : {"c", "e", "j"}, "F" : {"b", "f", "j"}, "G" : {"a", "g", "l"}, "H" : {"b", "d", "l"}, "I" : {"a", "e", "o"}, "J" : {"c", "f", "j"}, "K" : {"b", "h", "l"}, "L" : {"a", "i", "o"}, "M" : {"b", "f", "p"}, "N" : {"c", "e", "o"}, "O" : {"a", "g", "q"}, "P" : {"c", "f", "p"}, "Q" : {"a", "i", "q"} :{"a"}, "c":{"a", "b", "f"}, "e":{"d"}, "f":{"d", "e"}, "g":{"d","e"}, "h":{"d","e","f","g"}, "i":{"d","e","g"}, "l":{"j"}, "o":{"j","l"}, "p":{"j","l","o"}, "q":{"j","l","o","i"}, "a":set(), "d": set(), "j": set(), "p": set(), } allElementList = ["A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q…”

Section: } Allelementlist = ["unclassified

“…with the goal of describing and enumerating the sets A(λ, µ) as the weights λ and µ vary among the (dominant) fundamental weight lattice. Work to describe and enumerate the Weyl alternation sets has considered cases where λ is the highest root or a sum of the simple roots of a simple Lie algebra of types A r , B r , C r , and D r , see [2,4,[6][7][8][9]. The Weyl alternation sets of A 2 , A 3 , and all other simple Lie algebras of rank 2 have been described in [11,12].…”

Section: Introductionmentioning

confidence: 99%

On Kostant's weight $q$-multiplicity formula for $\mathfrak{sp}_6(\mathbb{C})$

Harris,

Hollander,

Qin

et al. 2021

Preprint

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Kostant's weight q-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. The q-analog of the partition function is a polynomial-valued function defined by ℘q(ξ) = k i=0 c i q i , where c i is the number of ways the weight ξ can be written as a sum of exactly i positive roots of a Lie algebra g. The evaluation of the q-multiplicity formula at q = 1 recovers the multiplicity of a weight in an irreducible highest weight representation of g. In this paper, we specialize to the Lie algebra sp 6 (C) and we provide a closed formula for the q-analog of Kostant's partition function, which extends recent results of Shahi, Refaghat, and Marefat. We also describe the supporting sets of the multiplicity formula (known as the Weyl alternation sets of sp 6 (C)), and use these results to provide a closed formula for the q-multiplicity for any pair of dominant integral weights of sp 6 (C). Throughout this work, we provide code to facilitate these computations.α∈Φ + α with Φ + being a set of positive roots of g. The terms of the alternating sum in Equation (1.1) are values of Kostant's partition function, which we denote by ℘, and which counts the number of ways to express its input as a nonnegative integral linear combination of the positive roots in Φ + .A well-known generalization of Kostant's weight multiplicity formula, due to Lusztig [17], is known as the q-analog of Kostant's weight multiplicity formula, and it replaces the partition function ℘ with its qanalog, denoted ℘ q . The q-analog of the partition function is defined as follows: For a weight ξ, ℘ q is a polynomial-valued function:

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“…Previous work concerning Weyl alternation sets includes determining and enumerating A(α, 0) when α is the highest root of the simple Lie algebras [4,7] and A(λ, 0) when λ is the sum of the simple roots of the classical Lie algebras [2]. In these cases the size of the Weyl alternation sets are given by constant coefficient homogeneous relations.…”

Section: Introductionmentioning

confidence: 99%

Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie Algebras

Harris,

Loving,

Ramirez

et al. 2019

Preprint

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The multiplicity of a weight in a finite-dimensional irreducible representation of a simple Lie algebra g can be computed via Kostant's weight multiplicity formula. This formula consists of an alternating sum over the Weyl group (a finite group) and involves a partition function known as Kostant's partition function. Motivated by the observation that, in practice, most terms in the sum are zero, our main results describe the elements of the Weyl alternation sets. The Weyl alternation sets are subsets of the Weyl group which contributes nontrivially to the multiplicity of a weight in a highest weight representation of the Lie algebras so4(C), so5(C), sp 4 (C), and the exceptional Lie algebra g2. By taking a geometric approach, we extend the work of Harris, Lescinsky, and Mabie on sl3(C), to provide visualizations of these Weyl alternation sets for all pairs of integral weights λ and µ of the Lie algebras considered.

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Kostant’s weight multiplicity formula and the Fibonacci and Lucas numbers

Cited by 8 publications

References 0 publications

When is the $q$-Multiplicity of a Weight a Power of $q$?

When is the $q$-Multiplicity of a Weight a Power of $q$?

On Kostant's weight $q$-multiplicity formula for $\mathfrak{sp}_6(\mathbb{C})$

Visualizing the Support of Kostant's Weight Multiplicity Formula for the Rank Two Lie Algebras

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