Dear Dan,
I suppose it is easiest for everyone to add it to the upcoming patch.
Ok!
What is the combinatorial R-matrix?
The combinatorial R-matrix is an affine crystal isomorphisms
R: K \otimes L \to L \otimes K
where K and L are Kirillov-Reshetikhin crystals. It maps
u_K \otimes u_L \mapsto u_L \otimes u_K
where u_K is the unique crystal elements of weight
s \Lambda_r - s*c*\Lambda_0 if K = B^{r,s}.
With the patch crystal-energy-as.patch on the combinat server
(this might change soon though, since I am going to fold the various
crystal patches once I know everything works)
sage: K = KirillovReshetikhinCrystal(['A',2,1],1,1)
sage: L = KirillovReshetikhinCrystal(['A',2,1],1,2)
sage: f = K.R_matrix(L)
sage: [[b,f(b)] for b in TensorProductOfCrystals(K,L)]
[[[[[1]], [[1, 1]]], [[[1, 1]], [[1]]]],
[[[[1]], [[1, 2]]], [[[1, 1]], [[2]]]],
[[[[1]], [[2, 2]]], [[[1, 2]], [[2]]]],
[[[[1]], [[1, 3]]], [[[1, 1]], [[3]]]],
[[[[1]], [[2, 3]]], [[[1, 2]], [[3]]]],
[[[[1]], [[3, 3]]], [[[1, 3]], [[3]]]],
[[[[2]], [[1, 1]]], [[[1, 2]], [[1]]]],
[[[[2]], [[1, 2]]], [[[2, 2]], [[1]]]],
[[[[2]], [[2, 2]]], [[[2, 2]], [[2]]]],
[[[[2]], [[1, 3]]], [[[2, 3]], [[1]]]],
[[[[2]], [[2, 3]]], [[[2, 2]], [[3]]]],
[[[[2]], [[3, 3]]], [[[2, 3]], [[3]]]],
[[[[3]], [[1, 1]]], [[[1, 3]], [[1]]]],
[[[[3]], [[1, 2]]], [[[1, 3]], [[2]]]],
[[[[3]], [[2, 2]]], [[[2, 3]], [[2]]]],
[[[[3]], [[1, 3]]], [[[3, 3]], [[1]]]],
[[[[3]], [[2, 3]]], [[[3, 3]], [[2]]]],
[[[[3]], [[3, 3]]], [[[3, 3]], [[3]]]]]
sage: K = KirillovReshetikhinCrystal(['D',4,1],1,1)
sage: L = KirillovReshetikhinCrystal(['D',4,1],2,1)
sage: f = K.R_matrix(L)
sage: T = TensorProductOfCrystals(K,L)
sage: b = T( K(rows=[[1]]), L(rows=[]) )
sage: f(b)
[[[2], [-2]], [[1]]]
Best wishes,
Anne
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