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Representation Theory

Published by the American Mathematical Society, the Representation Theory (ERT) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.7.

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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 PDF
Represent. Theory 7 (2003), 101-163 Request permission


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
  • Email:
  • 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:
  • Mark Shimozono
  • Affiliation: Department of Mathematics, 460 McBryde Hall, Virginia Tech, Blacksburg, Virginia 24061-0123
  • Email:
  • 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:
  • MathSciNet review: 1973369