Hi Nicolas,
>
> 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?
>
Yes you can (assuming you have 1 vector for each path in the crystal). It
is essentially by construction from the Verma module with the PBW basis but
thinking of the vectors f_* v. Although I think the good way to do this is
by the "adapted strings" method that de Graaf attributes to Littelmann in
his "Constructing canonical bases of quantized enveloping algebras". I am
pretty sure you can also tease this out directly from the definition of the
Kashiwara operators.
>
> 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?
>
>
So if you have a polarization and a basis for your submodule, you can use
that to do your projection. In particular, the crystal basis is the set b
\in V such that (b, b')_0 = \delta_{m,m'}, where m is the index of the
element b. However, that assumes you are working with coefficients in the
0-regular (functions f/g such that g(0) \neq 0) sublattice.
The more brute-force way of doing this is to construct a basis for
HV_{\lambda} using the gl_r action and extend that to V_{\lambda} using a
Gram-Schmidt-type approach to get a full basis of V_{\lambda} that
naturally has the projection, then conjugating that natural projection
matrix with the corresponding COB matrix to obtain your projection.
(I've done similar computations to compute highest weight elements, and I
found that stacking the e matrices and taking the kernel of that to be much
faster than taking the intersections of the kernels in Sage.)
>
> In general, refs about this type of computations in Lie algebras are
> very welcome ...
>
> de Graaf is the person who has done the most with these types of
computations (a number of which are implemented in GAP4).
http://www.science.unitn.it/~degraaf/pub.html
This paper of his
<https://www.researchgate.net/profile/Willem_De_Graaf/publication/46659882_Five_constructions_of_representations_of_quantum_groups/links/0c96053731dad5586a000000/Five-constructions-of-representations-of-quantum-groups.pdf>
is probably a good starting point as well.
Best,
Travis
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