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Representation Theory

Published by the American Mathematical Society since 1997, this electronic-only journal is devoted to research in representation theory and seeks to maintain a high standard for exposition as well as for mathematical content. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4165

The 2020 MCQ for Representation Theory is 0.71.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


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
Published electronically: March 4, 2003


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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Bibliographic 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