We establish a bijection between the set of rigged configurations and the set of tensor products of Kirillov–Reshetikhin crystals of type Dn(1) in full generality. We prove the invariance of rigged configurations under the action of the combinatorial R-matrix on tensor products and show that the bijection preserves certain statistics (cocharge and energy). As a result, we establish the fermionic formula for type Dn(1). In addition, we establish that the bijection is a classical crystal isomorphism.
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This work benefited from computations in SageMath [, ] (using implementations of crystals and rigged configurations by Schilling and Scrimshaw) and Mathematica (using an implementation of rigged configurations by Sakamoto ). AS and TS would like to thank Osaka City University for kind hospitality during their stay in July 2015. Both authors were partially supported by the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers “Mathematical Science of Symmetry, Topology and Moduli, Evolution of International Research Network based on OCAMI.” MO was partially supported by the Grants-in-Aid for Scientific Research No. 23340007 and No. 15K13429 from JSPS. The work of RS was partially supported by Grants-in-Aid for Scientific Research No. 25800026 from JSPS. AS was partially supported by NSF grants OCI–1147247 and DMS–1500050. TS was partially supported by NSF Grant OCI–1147247 and RTG Grant NSF/DMS–1148634.
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- Crystal bases
- Fermionic formula
- Kyoto path model
- Rigged configurations