The polynomial behavior of weight multiplicities for classical simple Lie algebras and classical affine Kac-Moody algebras

Benkart, Georgia; Kang, Seok-Jίn; Lee, Hyeonmi; Shin, Dong Yun

doi:10.1090/conm/248/03815

Cited by 8 publications

(31 citation statements)

References 3 publications

(4 reference statements)

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“…Given an ordered pair of partitions ( , ), the convention now [1,2] is to let the number of columns of height i in be the coefficient of i . Thus and encode information about the coefficients at the two ends of the Dynkin diagram of A n .…”

Section: Double-headed Weightsmentioning

confidence: 99%

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Tensor product stabilization in Kac–Moody algebras

Kleber

Viswanath

2006

Advances in Mathematics

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We consider a large class of series of symmetrizable Kac-Moody algebras (generically denoted X n ). This includes the classical series A n as well as others like E n whose members are of Indefinite type. The focus is to analyze the behavior of representations in the limit n → ∞. Motivated by the classical theory of A n = sl n+1 C, we consider tensor product decompositions of irreducible highest weight representations of X n and study how these vary with n. The notion of "double-headed" dominant weights is introduced. For such weights, we show that tensor product decompositions in X n do stabilize, generalizing the classical results for A n . The main tool used is Littelmann's celebrated path model. One can also use the stable multiplicities as structure constants to define a multiplication operation on a suitable space. We define this so-called stable representation ring and show that the multiplication operation is associative.

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Section: Double-headed Weightsmentioning

confidence: 99%

“…Such "double-headed" weights have been previously considered in Refs. [1][2][3]8,9] in the context of A n . Let H + 2 denote the set of ordered pairs of partitions (this definition will be slightly modified in the body of this paper).…”

Section: Introductionmentioning

confidence: 99%

Tensor product stabilization in Kac–Moody algebras

Kleber

Viswanath

2006

Advances in Mathematics

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“…n ) for all large n. They show that such γ's must have the form as in Theorem 1 (see proposition 1.22 of [1]). …”

Section: Bcd Diagramsmentioning

confidence: 89%

“…So for A (1) n we obtain stabilization of c ν λµ (n) and b λβ (n) for all λ, µ, ν, β, subject to a compatibility condition on their levels.…”

Section: Bcd Diagramsmentioning

confidence: 99%

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Dynkin Diagram Sequences and Stabilization Phenomena

Viswanath

2006

Communications in Algebra

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Abstract. We continue the study of stabilization phenomena for Dynkin diagram sequences initiated in the earlier work of Kleber and the present author. We consider a more general class of sequences than that of this earlier work, and isolate a condition on the weights that gives stabilization of tensor product and branching multiplicities. We show that all the results of the previous article can be naturally generalized to this setting. We also prove some properties of the partially ordered set of dominant weights of indefinite Kac-Moody algebras, and use this to give a more concrete definition of a stable representation ring. Finally, we consider the classical sequences Bn, Cn, Dn that fall outside the purview of the earlier work, and work out some easy-to-describe conditions on the weights which imply stabilization.

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A vector partition function for the multiplicities of slkC

2004

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We use Gelfand-Tsetlin diagrams to write down the weight multiplicity function for the Lie algebra sl k C (type A k−1 ) as a single partition function. This allows us to apply known results about partition functions to derive interesting properties of the weight diagrams. We relate this description to that of the Duistermaat-Heckman measure from symplectic geometry, which gives a large-scale limit way to look at multiplicity diagrams. We also provide an explanation for why the weight polynomials in the boundary regions of the weight diagrams exhibit a number of linear factors. Using symplectic geometry, we prove that the partition of the permutahedron into domains of polynomiality of the Duistermaat-Heckman function is the same as that for the weight multiplicity function, and give an elementary proof of this for sl 4 C (A 3 ).

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The polynomial behavior of weight multiplicities for classical simple Lie algebras and classical affine Kac-Moody algebras

Cited by 8 publications

References 3 publications

Tensor product stabilization in Kac–Moody algebras

Tensor product stabilization in Kac–Moody algebras

Dynkin Diagram Sequences and Stabilization Phenomena

A vector partition function for the multiplicities of slkC

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