#5879: [with patch, positive review] Added crystal of letters for type E
---------------------------+------------------------------------------------
Reporter: aschilling | Owner: aschilling
Type: enhancement | Status: new
Priority: major | Milestone: sage-4.0
Component: combinatorics | Keywords: combinat, crystals
---------------------------+------------------------------------------------
Comment(by bump):
The revision of the patch implementing concise representation is important
since
for tensor products of crystals the names of elements would be very
cumbersome.
The crystal at hand has highest weight the first fundamental weight,
lambda1 in the
Bourbaki notation. The dual crystal has highest weight lambda6:
{{{
O 2
|
|
O---O---O---O---O
1 3 4 5 6
E6
}}}
These both have degree 27. The adjoint representation has highest weight
lambda2
has degree 78 and can be constructed as follows:
{{{
sage: C = CrystalOfLetters(['E',6], element_print_style = 'compact')
sage: D = CrystalOfLetters(['E',6], element_print_style = 'compact',
dual=True)
sage: hwv=TensorProductOfCrystals(C,D).highest_weight_vectors(); hwv
[[+, -], [j, -], [x, -]]
sage: T = TensorProductOfCrystals(C,D,generators=[hwv[1]])
sage: T.cardinality()
78
sage: T.latex_file("/home/bump/tmp/e6-78.tex")
}}}
The last step assumes you have dot2tex installed. (Why isn't this
bundled with SAGE?)
Here is the resulting crystal of the adjoint representation:
http://sporadic.stanford.edu/bump/xtal/e6-78.pdf
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5879#comment:5>
Sage <http://sagemath.org/>
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