AUTHORS:
Bases: sage.structure.sage_object.SageObject, sage.structure.unique_representation.UniqueRepresentation
A Cartan type realized from a (Dynkin) diagram folding.
Given a Cartan type , we say
is a folded Cartan
type of
if there exists a diagram folding of the Dynkin
diagram of
onto
.
A folding of a simply-laced Dynkin diagram with index set
is an
automorphism
of
where all nodes any orbit of
are not
connected. The resulting Dynkin diagram
is induced by
where we identify edges in
which are not incident
and add a
-edge if we identify
incident edges and the arrow is
pointing towards the indicent note. We denote the index set of
by
, and by abuse of notation, we denote the folding by
.
We also have scaling factors for
and defined
as the unique numbers such that the map
is the smallest proper embedding of the weight lattice of
to
.
If the Cartan type is simply laced, the default folding is the one
induced from the identity map on .
If is affine type, the default embeddings we consider here are:
and were chosen based on virtual crystals. In particular, the diagram
foldings extend to crystal morphisms and gives a realization of
Kirillov-Reshetikhin crystals for non-simply-laced types as simply-laced
types. See [OSShimo03] and [FOS09] for more details. Here we can compute
where
are the translation factors
of the root system. In a more type-dependent way, we can define
as follows:
We note that only depends upon
.
If the Cartan type is finite, then we consider the classical foldings/embeddings induced by the above affine foldings/embeddings:
For more information on Cartan types, see sage.combinat.root_system.cartan_type.
Other foldings may be constructed by passing in an optional folding_of second argument. See below.
INPUT:
Note
If is an affine type, we assume the special node is fixed
under
.
EXAMPLES:
sage: fct = CartanType(['C',4,1]).as_folding(); fct
['C', 4, 1] as a folding of ['A', 7, 1]
sage: fct.scaling_factors()
Finite family {0: 2, 1: 1, 2: 1, 3: 1, 4: 2}
sage: fct.folding_orbit()
Finite family {0: (0,), 1: (1, 7), 2: (2, 6), 3: (3, 5), 4: (4,)}
A simply laced Cartan type can be considered as a virtual type of itself:
sage: fct = CartanType(['A',4,1]).as_folding(); fct
['A', 4, 1] as a folding of ['A', 4, 1]
sage: fct.scaling_factors()
Finite family {0: 1, 1: 1, 2: 1, 3: 1, 4: 1}
sage: fct.folding_orbit()
Finite family {0: (0,), 1: (1,), 2: (2,), 3: (3,), 4: (4,)}
Finite types:
sage: fct = CartanType(['C',4]).as_folding(); fct
['C', 4] as a folding of ['A', 7]
sage: fct.scaling_factors()
Finite family {1: 1, 2: 1, 3: 1, 4: 2}
sage: fct.folding_orbit()
Finite family {1: (1, 7), 2: (2, 6), 3: (3, 5), 4: (4,)}
sage: fct = CartanType(['F',4]).dual().as_folding(); fct
['F', 4] relabelled by {1: 4, 2: 3, 3: 2, 4: 1} as a folding of ['E', 6]
sage: fct.scaling_factors()
Finite family {1: 1, 2: 1, 3: 2, 4: 2}
sage: fct.folding_orbit()
Finite family {1: (1, 6), 2: (3, 5), 3: (4,), 4: (2,)}
REFERENCES:
[OSShimo03] | M. Okado, A. Schilling, M. Shimozono.
“Virtual crystals and fermionic formulas for type ![]() ![]() ![]() |
Return the Cartan type of self.
EXAMPLES:
sage: fct = CartanType(['C', 4, 1]).as_folding()
sage: fct.cartan_type()
['C', 4, 1]
Return the Cartan type of the virtual space.
EXAMPLES:
sage: fct = CartanType(['C', 4, 1]).as_folding()
sage: fct.folding_of()
['A', 7, 1]
Return the orbits under the automorphism as a
dictionary (of tuples).
EXAMPLES:
sage: fct = CartanType(['C', 4, 1]).as_folding()
sage: fct.folding_orbit()
Finite family {0: (0,), 1: (1, 7), 2: (2, 6), 3: (3, 5), 4: (4,)}
Return the scaling factors of self.
EXAMPLES:
sage: fct = CartanType(['C', 4, 1]).as_folding()
sage: fct.scaling_factors()
Finite family {0: 2, 1: 1, 2: 1, 3: 1, 4: 2}
sage: fct = CartanType(['BC', 4, 2]).as_folding()
sage: fct.scaling_factors()
Finite family {0: 1, 1: 1, 2: 1, 3: 1, 4: 2}
sage: fct = CartanType(['BC', 4, 2]).dual().as_folding()
sage: fct.scaling_factors()
Finite family {0: 2, 1: 1, 2: 1, 3: 1, 4: 1}
sage: CartanType(['BC', 4, 2]).relabel({0:4, 1:3, 2:2, 3:1, 4:0}).as_folding().scaling_factors()
Finite family {0: 2, 1: 1, 2: 1, 3: 1, 4: 1}