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Virtual crystals and fermionic formulas of type $D_{n+1}^{(2)}$, $A_{2n}^{(2)}$, and $C_n^{(1)}$

Authors: Masato Okado, Anne Schilling and Mark Shimozono
Journal: Represent. Theory 7 (2003), 101-163
MSC (2000): Primary 81R50, 81R10, 17B37; Secondary 05A30, 82B23
Published electronically: March 4, 2003
MathSciNet review: 1973369
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Abstract: We introduce “virtual” crystals of the affine types $\mathfrak {g}=D_{n+1}^{(2)}$, $A_{2n}^{(2)}$ and $C_n^{(1)}$ by naturally extending embeddings of crystals of types $B_n$ and $C_n$ into crystals of type $A_{2n-1}$. Conjecturally, these virtual crystals are the crystal bases of finite dimensional $U_q’(\mathfrak {g})$-modules associated with multiples of fundamental weights. We provide evidence and in some cases proofs of this conjecture. Recently, fermionic formulas for the one-dimensional configuration sums associated with tensor products of the finite dimensional $U_q’(\mathfrak {g})$-modules were conjectured by Hatayama et al. We provide proofs of these conjectures in specific cases by exploiting duality properties of crystals and rigged configuration techniques. For type $A_{2n}^{(2)}$ we also conjecture a new fermionic formula coming from a different labeling of the Dynkin diagram.

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Additional Information

Masato Okado
Affiliation: Department of Informatics and Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan

Anne Schilling
Affiliation: Department of Mathematics, University of California, One Shields Avenue, Davis, California 95616-8633
MR Author ID: 352840
ORCID: 0000-0002-2601-7340

Mark Shimozono
Affiliation: Department of Mathematics, 460 McBryde Hall, Virginia Tech, Blacksburg, Virginia 24061-0123

Keywords: Crystal bases, quantum affine Lie algebras, fermionic formulas, rigged configurations, contragredient duality
Received by editor(s): January 14, 2002
Received by editor(s) in revised form: November 27, 2002
Published electronically: March 4, 2003
Additional Notes: The third author was partially supported by NSF grant DMS-9800941
Article copyright: © Copyright 2003 American Mathematical Society