Virtual crystals and fermionic formulas of type $D_{n+1}^{(2)}$, $A_{2n}^{(2)}$, and $C_n^{(1)}$
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- by Masato Okado, Anne Schilling and Mark Shimozono
- Represent. Theory 7 (2003), 101-163
- DOI: https://doi.org/10.1090/S1088-4165-03-00155-9
- Published electronically: March 4, 2003
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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.References
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Bibliographic Information
- Masato Okado
- Affiliation: Department of Informatics and Mathematical Science, Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan
- Email: okado@sigmath.es.osaka-u.ac.jp
- 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
- Email: anne@math.ucdavis.edu
- Mark Shimozono
- Affiliation: Department of Mathematics, 460 McBryde Hall, Virginia Tech, Blacksburg, Virginia 24061-0123
- Email: mshimo@math.vt.edu
- 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
- © Copyright 2003 American Mathematical Society
- Journal: Represent. Theory 7 (2003), 101-163
- MSC (2000): Primary 81R50, 81R10, 17B37; Secondary 05A30, 82B23
- DOI: https://doi.org/10.1090/S1088-4165-03-00155-9
- MathSciNet review: 1973369