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()