# Bijection classes for type $$D_n^{(1)}$$¶

Part of the (internal) classes which runs the bijection between rigged configurations and KR tableaux of type $$D_n^{(1)}$$.

AUTHORS:

• Travis Scrimshaw (2011-04-15): Initial version
class sage.combinat.rigged_configurations.bij_type_D.KRTToRCBijectionTypeD(tp_krt)

Specific implementation of the bijection from KR tableaux to rigged configurations for type $$D_n^{(1)}$$.

This inherits from type $$A_n^{(1)}$$ because we use the same methods in some places.

doubling_map()

Perform the doubling map of the rigged configuration at the current state of the bijection.

This is the map $$B(\Lambda) \hookrightarrow B(2 \Lambda)$$ which doubles each of the rigged partitions and updates the vacancy numbers accordingly.

halving_map()

Perform the halving map of the rigged configuration at the current state of the bijection.

This is the inverse map to $$B(\Lambda) \hookrightarrow B(2 \Lambda)$$ which halves each of the rigged partitions and updates the vacancy numbers accordingly.

next_state(val)

Build the next state for type $$D_n^{(1)}$$.

run(verbose=False)

Run the bijection from a tensor product of KR tableaux to a rigged configuration for type $$D_n^{(1)}$$.

INPUT:

• tp_krt – A tensor product of KR tableaux
• verbose – (Default: False) Display each step in the bijection

EXAMPLES:

sage: KRT = crystals.TensorProductOfKirillovReshetikhinTableaux(['D', 4, 1], [[2,1]])
sage: from sage.combinat.rigged_configurations.bij_type_D import KRTToRCBijectionTypeD
sage: KRTToRCBijectionTypeD(KRT(pathlist=[[-3,2]])).run()

-1[ ]-1

2[ ]2

-1[ ]-1

-1[ ]-1

class sage.combinat.rigged_configurations.bij_type_D.RCToKRTBijectionTypeD(RC_element)

Specific implementation of the bijection from rigged configurations to tensor products of KR tableaux for type $$D_n^{(1)}$$.

doubling_map()

Perform the doubling map of the rigged configuration at the current state of the bijection.

This is the map $$B(\Lambda) \hookrightarrow B(2 \Lambda)$$ which doubles each of the rigged partitions and updates the vacancy numbers accordingly.

halving_map()

Perform the halving map of the rigged configuration at the current state of the bijection.

This is the inverse map to $$B(\Lambda) \hookrightarrow B(2 \Lambda)$$ which halves each of the rigged partitions and updates the vacancy numbers accordingly.

next_state(height)

Build the next state for type $$D_n^{(1)}$$.

run(verbose=False, build_graph=False)

Run the bijection from rigged configurations to tensor product of KR tableaux for type $$D_n^{(1)}$$.

INPUT:

• verbose – (default: False) display each step in the bijection
• build_graph – (default: False) build the graph of each step of the bijection

EXAMPLES:

sage: RC = RiggedConfigurations(['D', 4, 1], [[2, 1]])
sage: x = RC(partition_list=[[1],[1],[1],[1]])
sage: from sage.combinat.rigged_configurations.bij_type_D import RCToKRTBijectionTypeD
sage: RCToKRTBijectionTypeD(x).run()
[[2], [-3]]
sage: bij = RCToKRTBijectionTypeD(x)
sage: bij.run(build_graph=True)
[[2], [-3]]
sage: bij._graph
Digraph on 3 vertices