Computing Multiplicities of Lie Group Representations

Abstract

For fixed compact connected Lie groups H \subseteq G, we provide a polynomial time algorithm to compute the multiplicity of a given irreducible representation of H in the restriction of an irreducible representation of G. Our algorithm is based on a finite difference formula which makes the multiplicities amenable to Barvinok's algorithm for counting integral points in polytopes. The Kronecker coefficients of the symmetric group, which can be seen to be a special case of such multiplicities, play an important role in the geometric complexity theory approach to the P vs. NP problem. Whereas their computation is known to be #P-hard for Young diagrams with an arbitrary number of rows, our algorithm computes them in polynomial time if the number of rows is bounded. We complement our work by showing that information on the asymptotic growth rates of multiplicities in the coordinate rings of orbit closures does not directly lead to new complexity-theoretic obstructions beyond what can be obtained from the moment polytopes of the orbit closures. Non-asymptotic information on the multiplicities, such as provided by our algorithm, may therefore be essential in order to find obstructions in geometric complexity theory.

10 pages

References in corpus (10)

• Quantum marginal problem and N-representability
• Estimating the spectrum of a density operator
• The Spectra of Density Operators and the Kronecker Coefficients of the Symmetric Group
• On Nonzero Kronecker Coefficients and their Consequences for Spectra
• Quantum state transformations and the Schubert calculus
• Nonvanishing of Kronecker coefficients for rectangular shapes
• Even Partitions in Plethysms
• On the complexity of computing Kronecker coefficients
• Highly Entangled States With Almost No Secrecy
• Chopped and sliced cones and representations of Kac-Moody algebras

Cited by in corpus (7)

• On vanishing of Kronecker coefficients
• Tensor Models, Kronecker coefficients and Permutation Centralizer Algebras
• Eigenvalue Distributions of Reduced Density Matrices
• Plethysm and lattice point counting
• Rectangular symmetries for coefficients of symmetric functions
• All Kronecker coefficients are reduced Kronecker coefficients
• Weight multiplicity formulas for bivariate representations of classical Lie algebras