Added crystal of letters for type E

[anneschilling]

This patch adds crystal of letters for type E corresponding to
the highest weight crystal B(\Lambda_1) and its dual B(\Lambda_6) (in the sage labeling convention of the Dynkin nodes).

This patch depends on ticket #5120.

CC: sage-combinat-commits

Component: combinatorics

Keywords: combinat, crystals

Issue created by migration from https://trac.sagemath.org/ticket/5879

[anneschilling]

Description changed:

--- 
+++ 
@@ -1,2 +1,4 @@
 This patch adds crystal of letters for type E corresponding to
 the highest weight crystal B(\Lambda_1) and its dual B(\Lambda_6) (in the sage labeling convention of the Dynkin nodes).
+
+This patch depends on ticket #5120.

[dwbump]

comment:2

For what it's worth, here's a picture of this crystal:

http://sporadic.stanford.edu/bump/xtal/e6.pdf

What is the logic of the representations of the elements? That is, why [-5,2,6]?

[dwbump]

comment:3

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.

[anneschilling]

comment:4

What is the logic of the representations of the elements? That is, why [-5,2,6]?

The nodes of this crystal are labeled by their weights. [-5,2,6] indicates that
a 5-arrow is coming in, and a 2-arrow and 6-arrow is leaving the vertex.

I now also added a __repr__ method that allows for concise printing of the nodes by
+abcdefghijklmnopqrstuvwxyz and -ABCDEFGHIJKLMNOPQRSTUVWXYZ in the dual crystal.

[dwbump]

comment:5

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

[sagetrac-mabshoff]

comment:6

Mhh, what is the credit situation here? "Brant Jones" was added to the list of authors, so is this shared credit for Anne and Brant for authorship?

Cheers,

Michael