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. Thiéry "Isil" <nthi...@users.sf.net>
http://Nicolas.Thiery.name/
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