Dear crystals and Lie algebra fans, I am doing computations in a Lie algebra representation V; actually a pretty simple one, though huge (in case this is relevant, V is a subspace of the ring of polynomials in r sets of n variables, under the action of gl_r. The nice feature is that weight spaces are directly given by the multigrading, and the e and f operators are given by polarization).
Question I: assume that I have a highest weight vector v for a simple submodule of V. Consider the crystal of tableaux of the same weight. Take a set S of tableaux T; for each of them, pick a string of e crystal operators going from the highest weight tableau to T. Can I by any chance assume that, if the corresponding strings of e operators in gl_r are applied on v, I get linearly independent elements in V? Question II: assume that I have a weight space V_\lambda; I can easily compute the subspace HV_\lambda of highest weight vectors in V_\lambda by taking the joint kernel of the e operators. But for my computation I would need to instead have a *projection* from V_\lambda to HV_\lambda. Is there a way to construct such a projection, e.g. in term of the operators of the Lie algebra? In general, refs about this type of computations in Lie algebras are very welcome ... Thanks in advance! Cheers, Nicolas -- Nicolas M. ThiƩry "Isil" <nthi...@users.sf.net> http://Nicolas.Thiery.name/ -- You received this message because you are subscribed to the Google Groups "sage-combinat-devel" group. To unsubscribe from this group and stop receiving emails from it, send an email to sage-combinat-devel+unsubscr...@googlegroups.com. To post to this group, send email to sage-combinat-devel@googlegroups.com. Visit this group at https://groups.google.com/group/sage-combinat-devel. For more options, visit https://groups.google.com/d/optout.