On 3/28/11 6:41 AM, Anne Schilling wrote:
On 3/28/11 6:28 AM, bump wrote:
Yes, it also exists for any Cartan type, but my impression was that then
it is called Lusztig involution. My plan is indeed to eventually implement
the Lusztig involution in CrystalOfTableauxElements. What do you think?
That sounds like the right approach. Actually the Lusztig involution
would be pretty easy to
implement for irreducible crystals and would give a way of checking
the
correctness of the patch in question.
I don't have the details on top of my head, so I'll just speak vaguely
here. Unless the algorithms are basically identical, it seems very
reasonable to have a generic (type-free?) implementation in
CrystalOfTableauxElements using the crystal structure, and a
specialized implementation in Tableaux using the combinatorics of
tableaux. And to include some tests that compare the results and check
their consistency!
I am agreeing with this. For crystals, the algorithm is as follows.
There is an involution of the indices which for type A is i --> r+1-i
but
for other types may be trivial. Anyway call this i -> i'. Now
if v is in the crystal C, find a path from the highest weight vector
to v.
Then start with the lowest weight vector and find the symmetrical
path to some vertex, where the symmetry replaces e(i) -> f(i').
Where you end up is the involution of v.
Yes, indeed! In fact you can obtain the map i-> i' by
alpha_i -> alpha_{i'} := -w_0(alpha_i) where w_0 is the longest
element in the corresponding Weyl group. It is the identity except for
type A_n.
Hugh pointed out to me that the map i->i' is of course also not the
identity for type D_n, n odd (and exceptional types, but they are not
implemented in CrystalOfTableaux).
I can add the Lusztig involution on crystal elements to this patch if it
is desired and add tests to check that for type A it gives the same as
the Schuetzenberger involution.
I just posted a new version of the patch on trac and the sage-combinat queue
with the Lusztig involution implemented. It is tested that it gives the
same result as the Schuetzenberger involution for type A.
Cheers,
Anne
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