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:

There exists a unique arrow (multiple bond) in .
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 $\gamma_i = 1$ .
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:

Wikipedia article Dynkin_diagram#Folding

[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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