On the Computation of Clebsch-Gordan Coefficients and the Dilation\n Effect

2005/01/25 by Jesús A. De Loera, De Loera, Jesús A., Tyrrell B. McAllister +1 · 2 citations
Mathematics · #17B10 (Primary) 68R05 (Secondary) #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.math/0501446

openalex publication_date 2005/01/25 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

We investigate the problem of computing tensor product multiplicities for\ncomplex semisimple Lie algebras. Even though computing these numbers is #P-hard\nin general, we show that if the rank of the Lie algebra is assumed fixed, then\nthere is a polynomial time algorithm, based on counting the lattice points in\npolytopes. In fact, for Lie algebras of type A_r, there is an algorithm, based\non the ellipsoid algorithm, to decide when the coefficients are nonzero in\npolynomial time for arbitrary rank. Our experiments show that the lattice point\nalgorithm is superior in practice to the standard techniques for computing\nmultiplicities when the weights have large entries but small rank. Using an\nimplementation of this algorithm, we provide experimental evidence for\nconjectured generalizations of the saturation property of\nLittlewood--Richardson coefficients. One of these conjectures seems to be valid\nfor types B_n, C_n, and D_n.\n

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• On Vanishing of Gromov--Witten Invariants

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