Root system data for folded Cartan types

AUTHORS:

  • Travis Scrimshaw (2013-01-12) - Initial version
class sage.combinat.root_system.type_folded.CartanTypeFolded(cartan_type, folding_of, orbit)

Bases: sage.structure.sage_object.SageObject, sage.structure.unique_representation.UniqueRepresentation

A Cartan type realized from a (Dynkin) diagram folding.

Given a Cartan type X, we say \hat{X} is a folded Cartan type of X if there exists a diagram folding of the Dynkin diagram of \hat{X} onto X.

A folding of a simply-laced Dynkin diagram D with index set I is an automorphism \sigma of D where all nodes any orbit of \sigma are not connected. The resulting Dynkin diagram \hat{D} is induced by I / \sigma where we identify edges in \hat{D} which are not incident and add a k-edge if we identify k incident edges and the arrow is pointing towards the indicent note. We denote the index set of \hat{D} by \hat{I}, and by abuse of notation, we denote the folding by \sigma.

We also have scaling factors \gamma_i for i \in \hat{I} and defined as the unique numbers such that the map \Lambda_j \mapsto \gamma_j \sum_{i \in \sigma^{-1}(j)} \Lambda_i is the smallest proper embedding of the weight lattice of X to \hat{X}.

If the Cartan type is simply laced, the default folding is the one induced from the identity map on D.

If X is affine type, the default embeddings we consider here are:

\begin{array}{ccl}
C_n^{(1)}, A_{2n}^{(2)}, A_{2n}^{(2)\dagger}, D_{n+1}^{(2)}
& \hookrightarrow & A_{2n-1}^{(1)}, \\
A_{2n-1}^{(2)}, B_n^{(1)} & \hookrightarrow & D_{n+1}^{(1)}, \\
E_6^{(2)}, F_4^{(1)} & \hookrightarrow & E_6^{(1)}, \\
D_4^{(3)}, G_2^{(1)} & \hookrightarrow & D_4^{(1)},
\end{array}

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 \gamma_i = \max(c) / c_i where (c_i)_i are the translation factors of the root system. In a more type-dependent way, we can define \gamma_i as follows:

  1. There exists a unique arrow (multiple bond) in X.
    1. Suppose the arrow points towards 0. Then \gamma_i = 1 for all i \in I.
    2. Otherwise \gamma_i is the order of \sigma for all i in the connected component of 0 after removing the arrow, else \gamma_i = 1.
  2. There is not a unique arrow. Thus \hat{X} = A_{2n-1}^{(1)} and \gamma_i = 1 for all 1 \leq i \leq n-1. If i \in \{0, n\}, then \gamma_i = 2 if the arrow incident to i points away and is 1 otherwise.

We note that \gamma_i only depends upon X.

If the Cartan type is finite, then we consider the classical foldings/embeddings induced by the above affine foldings/embeddings:

\begin{aligned}
C_n & \hookrightarrow A_{2n-1}, \\
B_n & \hookrightarrow D_{n+1}, \\
F_4 & \hookrightarrow E_6, \\
G_2 & \hookrightarrow D_4.
\end{aligned}

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:

  • cartan_type – the Cartan type X to create the folded type
  • folding_of – the Cartan type \hat{X} which X is a folding of
  • orbit – the orbit of the Dynkin diagram automorphism \sigma given as a list of lists where the a-th list corresponds to the a-th entry in I or a dictionary with keys in I and values as lists

Note

If X is an affine type, we assume the special node is fixed under \sigma.

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 D_{n+1}^{(2)}, A_{2n}^{(2)}, and C_n^{(1)}”. Representation Theory. 7 (2003). 101-163. doi:10.1.1.192.2095, Arxiv 0810.5067.
cartan_type()

Return the Cartan type of self.

EXAMPLES:

sage: fct = CartanType(['C', 4, 1]).as_folding()
sage: fct.cartan_type()
['C', 4, 1]
folding_of()

Return the Cartan type of the virtual space.

EXAMPLES:

sage: fct = CartanType(['C', 4, 1]).as_folding()
sage: fct.folding_of()
['A', 7, 1]
folding_orbit()

Return the orbits under the automorphism \sigma 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,)}
scaling_factors()

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}

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