Rigged Configurations of \(\mathcal{B}(\infty)\)¶
AUTHORS:
- Travis Scrimshaw (2013-04-16): Initial version
-
sage.combinat.rigged_configurations.rc_infinity.
InfinityCrystalOfNonSimplyLacedRC
¶ Rigged configurations for \(\mathcal{B}(\infty)\) in non-simply-laced types.
-
sage.combinat.rigged_configurations.rc_infinity.
InfinityCrystalOfRiggedConfigurations
¶ Rigged configuration model for \(\mathcal{B}(\infty)\).
The crystal is generated by the empty rigged configuration with the same crystal structure given by the
highest weight model
except we remove the condition that the resulting rigged configuration needs to be valid when applying \(f_a\).INPUT:
cartan_type
– a Cartan type
EXAMPLES:
For simplicity, we display all of the rigged configurations horizontally:
sage: RiggedConfigurations.options(display='horizontal')
We begin with a simply-laced finite type:
sage: RC = crystals.infinity.RiggedConfigurations(['A', 3]); RC The infinity crystal of rigged configurations of type ['A', 3] sage: RC.options(display='horizontal') sage: mg = RC.highest_weight_vector(); mg (/) (/) (/) sage: elt = mg.f_string([2,1,3,2]); elt 0[ ]0 -2[ ]-1 0[ ]0 -2[ ]-1 sage: elt.e(1) sage: elt.e(3) sage: mg.f_string([2,1,3,2]).e(2) -1[ ]-1 0[ ]1 -1[ ]-1 sage: mg.f_string([2,3,2,1,3,2]) 0[ ]0 -3[ ][ ]-1 -1[ ][ ]-1 -2[ ]-1
Next we consider a non-simply-laced finite type:
sage: RC = crystals.infinity.RiggedConfigurations(['C', 3]) sage: mg = RC.highest_weight_vector() sage: mg.f_string([2,1,3,2]) 0[ ]0 -1[ ]0 0[ ]0 -1[ ]-1 sage: mg.f_string([2,3,2,1,3,2]) 0[ ]-1 -1[ ][ ]-1 -1[ ][ ]0 -1[ ]0
We can construct rigged configurations using a diagram folding of a simply-laced type. This yields an equivalent but distinct crystal:
sage: vct = CartanType(['C', 3]).as_folding() sage: VRC = crystals.infinity.RiggedConfigurations(vct) sage: mg = VRC.highest_weight_vector() sage: mg.f_string([2,1,3,2]) 0[ ]0 -2[ ]-1 0[ ]0 -2[ ]-1 sage: mg.f_string([2,3,2,1,3,2]) -1[ ]-1 -2[ ][ ][ ]-1 -1[ ][ ]0 sage: G = RC.subcrystal(max_depth=5).digraph() sage: VG = VRC.subcrystal(max_depth=5).digraph() sage: G.is_isomorphic(VG, edge_labels=True) True
We can also construct \(B(\infty)\) using rigged configurations in affine types:
sage: RC = crystals.infinity.RiggedConfigurations(['A', 3, 1]) sage: mg = RC.highest_weight_vector() sage: mg.f_string([0,1,2,3,0,1,3]) -1[ ]0 -1[ ]-1 1[ ]1 -1[ ][ ]-1 -1[ ]0 -1[ ]-1 sage: RC = crystals.infinity.RiggedConfigurations(['C', 3, 1]) sage: mg = RC.highest_weight_vector() sage: mg.f_string([1,2,3,0,1,2,3,3,0]) -2[ ][ ]-1 0[ ]1 0[ ]0 -4[ ][ ][ ]-2 0[ ]0 0[ ]-1 sage: RC = crystals.infinity.RiggedConfigurations(['A', 6, 2]) sage: mg = RC.highest_weight_vector() sage: mg.f_string([1,2,3,0,1,2,3,3,0]) 0[ ]-1 0[ ]1 0[ ]0 -4[ ][ ][ ]-2 0[ ]-1 0[ ]1 0[ ]-1
We reset the global options:
sage: RiggedConfigurations.options._reset()