An algorithm for computing weight multiplicities in irreducible modules\n for complex semisimple Lie algebras

2016/03/23 by Mikaël Cavallin, Cavallin, Mikaël · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Advanced Algebra and Geometry

paper · pdf · doi:10.48550/arxiv.1603.07196

Abstract

Let mathfrakg be a semisimple Lie algebra over \ℂ having rank\nl and let V=L(\λ) be an irreducible finite-dimensional\n mathfrakg-module having highest weight \λ. Computations of weight\nmultiplicities in V, usually based on Freudenthal's formula, are in general\ndifficult to carry out in large ranks or for \λ with large coefficients\n(in terms of the fundamental weights). In this paper, we first show that in\nsome situations, these coefficients can be "lowered" in order to simplify the\ncalculations. We then investigate how this can be used to improve the\naforementioned formula of Freudenthal, leading to a more efficient version of\nthe latter in terms of complexity as well as a way of dealing with certain\ncomputations in unbounded ranks. We conclude by illustrating the last assertion\nwith a concrete example.\n

Cited by

• Irreducible representations of simple Lie algebras with maximum weight multiplicity 2

Related

• Layer structure of irreducible Lie algebra modules
• A lower bound for the dimension of a highest weight module
• Weight modules of direct limit Lie algebras
• Weights of simple highest weight modules over a complex semisimple Lie algebra
• Computing weight q-multiplicities for the representations of the\n simple Lie algebras