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Isaac Newton Institute for Mathematical Sciences

Proper subalgebras of compact Lie algebras and features of Littlewood-Richardson coefficients

Presenter: Robert Zeier (Technische Universität München, Germany)

Co-author: Zoltán Zimborás (University of the Basque Country UPV/EHU, Spain)

Abstract

How can one distinguish a compact Lie algebra from its proper subalgebras? Classically, one would compute the Lie-algebra closures and compare the resulting dimensions. We obtain conditions which take advantage of symmetries of the involved Lie algebras and which can be directly decided on sets of generators. We extend the work of [1] where it is noted that the commutant of the tensor square of the defining representation of su(N) has dimension two and where it is shown that the corresponding commutant for a proper subalgebra of su(N) is at least three-dimensional. The proof in [1] applies representation-theoretic methods and relies critically on a classification of representations whose alternating square is irreducible [2]. Similar results can be obtained for orthogonal and unitary symplectic algebras (see also [3]). We analyse the general case and develop classification-free proof techniques which highlight curious features of (generalised) Littlewood-Richardson coefficients, such as given in [4]. These features also stress how the representation rings of compact connected Lie groups differ from the ones of finite groups.

[1] R. Zeier and T. Schulte-Herbrüggen, Symmetry principles in quantum systems theory, J. Math. Phys. 52:113510 (2011) [2] E. B. Dynkin, The maximal subgroups of the classical groups, Am. Math. Soc. Transl., Ser. 2, 6:245 (1957) [3] Z. Zimborás, R. Zeier, M. Keyl, and T. Schulte-Herbrüggen, A Dynamic Systems Approach to Fermions and Their Relation to Spins, http://arxiv.org/abs/1211.2226v3 [4] R. Coquereaux and J.-B. Zuber, On sums of tensor and fusion multiplicities, J. Phys. A 44:295208 (2011)