#5879: [with patch, favorable review] Added crystal of letters for type E
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Reporter: aschilling | Owner: aschilling
Type: enhancement | Status: new
Priority: major | Milestone: sage-3.4.2
Component: combinatorics | Keywords: combinat, crystals
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Comment(by bump):
After some testing I am confident that the crystal is correct.
Adding this one crystal (with its dual) actually gives access to
all E6 crystals (modulo computational complexity) since they can
be obtained as subcrystals of tensor products of these two. For
example, the second fundamental weight corresponds to a representation
of degree 351 whose crystal can be obtained thus:
{{{
sage: C = CrystalOfLetters("E6")
sage: T = TensorProductOfCrystals(C,C,generators=[[C[0],C[0]]])
sage: T.cardinality()
351
}}}
The fact that this process (using both crystals) is complete to give all
crystals is
confirmed here:
http://groups.google.com/group/sage-combinat-
devel/msg/8ef5b6b5b529e51b?hl=en
This crystal is also interesting because the action of the E6 Weyl group
on this
crystal of degree 27 must be equivalent to the action of E6 on the 27
lines on a cubic surface.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/5879#comment:3>
Sage <http://sagemath.org/>
Sage - Open Source Mathematical Software: Building the Car Instead of
Reinventing the Wheel
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