Affine Highest Weight Crystals

Affine highest weight crystals are infinite-dimensional. Hence, to work with them in Sage, we need some further tools.

Littelmann path model

The Littelmann path model for highest weight crystals is implemented in Sage. It models finite highest crystals as well as affine highest weight crystals which are infinite dimensional. The elements of the crystal are piecewise linear maps in the weight space. For more information on the Littelmann path model, see [L1995].

Since the affine highest weight crystals are infinite, it is not possible to list all elements or draw the entire crystal graph. However, if the user is only interested in the crystal up to a certain distance or depth from the highest weight element, then one can work with the corresponding subcrystal. To view the corresponding upper part of the crystal, one can build the associated digraph:

sage: R = RootSystem(['C',3,1])
sage: La = R.weight_space().basis()
sage: LS = crystals.LSPaths(2*La[1]); LS
The crystal of LS paths of type ['C', 3, 1] and weight 2*Lambda[1]
sage: C = LS.subcrystal(max_depth=3)
sage: sorted(C, key=str)
[(-Lambda[0] + 2*Lambda[1] - Lambda[2] + Lambda[3], Lambda[1]),
 (-Lambda[0] + Lambda[1] + Lambda[2], Lambda[0] - Lambda[1] + Lambda[2]),
 (-Lambda[0] + Lambda[1] + Lambda[2], Lambda[1]),
 (-Lambda[1] + 2*Lambda[2], Lambda[1]),
 (2*Lambda[0] - 2*Lambda[1] + 2*Lambda[2],),
 (2*Lambda[1],),
 (Lambda[0] + Lambda[2] - Lambda[3], Lambda[1]),
 (Lambda[0] - Lambda[1] + Lambda[2], Lambda[1]),
 (Lambda[0] - Lambda[2] + Lambda[3], Lambda[0] - Lambda[1] + Lambda[2]),
 (Lambda[0] - Lambda[2] + Lambda[3], Lambda[1])]

sage: G = LS.digraph(subset = C)
sage: view(G, pdflatex=True, tightpage=True)  #optional - dot2tex graphviz
../_images/LScrystal.png

The Littelmann path model also lends itself as a model for level zero crystals which are bi-infinite. To cut out a slice of these crystals, one can use the direction option in subcrystal:

sage: R = RootSystem(['A',2,1])
sage: La = R.weight_space(extended = True).basis()
sage: LS = crystals.LSPaths(La[1]-La[0]); LS
The crystal of LS paths of type ['A', 2, 1] and weight -Lambda[0] + Lambda[1]
sage: C = LS.subcrystal(max_depth=2, direction = 'both')
sage: G = LS.digraph(subset = C)
sage: G.set_latex_options(edge_options = lambda (u,v,label): ({}))
sage: view(G, pdflatex=True, tightpage=True)  #optional - dot2tex graphviz
../_images/level_zero_crystal.png

Modified Nakajima monomials

Modified Nakajima monomials have also been implemented in Sage and models highest weight crystals in all symmetrizable types. The elements are given in terms of commuting variables \(Y_i(n)\) where \(i \in I\) and \(n \in \ZZ_{\geq 0}\). For more information on the modified Nakajima monomials, see [KKS2007].

We give an example in affine type and verify that up to depth 3, it agrees with the Littelmann path model:

sage: La = RootSystem(['C',3,1]).weight_space().fundamental_weights()
sage: LS = crystals.LSPaths(2*La[1]+La[2])
sage: SL = LS.subcrystal(max_depth=3)
sage: GL = LS.digraph(subset=SL)

sage: La = RootSystem(['C',3,1]).weight_lattice().fundamental_weights()
sage: M = crystals.NakajimaMonomials(['C',3,1], 2*La[1]+La[2])
sage: SM = M.subcrystal(max_depth=3)
sage: GM = M.digraph(subset=SM)
sage: GL.is_isomorphic(GM, edge_labels=True)
True

Now we do an example of a simply-laced (so symmetrizable) hyperbolic type \(H_1^{(4)}\), which comes from the complete graph on 4 vertices:

sage: CM = CartanMatrix([[2, -1, -1,-1],[-1,2,-1,-1],[-1,-1,2,-1],[-1,-1,-1,2]]); CM
[ 2 -1 -1 -1]
[-1  2 -1 -1]
[-1 -1  2 -1]
[-1 -1 -1  2]
sage: La = RootSystem(CM).weight_lattice().fundamental_weights()
sage: M = crystals.NakajimaMonomials(CM, La[0])
sage: SM = M.subcrystal(max_depth=4)
sage: GM = M.digraph(subset=SM) # long time
../_images/hyperbolic_La0.png

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