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Crystal energy functions via the charge in types A and C

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Abstract

The Ram–Yip formula for Macdonald polynomials (at t = 0) provides a statistic which we call charge. In types A and C it can be defined on tensor products of Kashiwara–Nakashima single column crystals. In this paper we prove that the charge is equal to the (negative of the) energy function on affine crystals. The algorithm for computing charge is much simpler and can be more efficiently computed than the recursive definition of energy in terms of the combinatorial R-matrix.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, State University of New York at Albany, Albany, NY, 12222, USA

    Cristian Lenart

  2. Department of Mathematics, University of California, One Shields Avenue, Davis, CA, 95616-8633, USA

    Anne Schilling

Authors
  1. Cristian Lenart
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  2. Anne Schilling
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Correspondence to Cristian Lenart.

Additional information

C. Lenart was partially supported by the Research in Pairs program of Mathematiches Forschungsinstitut Oberwolfach, the NSF grant DMS-1101264, and an Individual Development Award from SUNY Albany. A. Schilling was partially supported by the NSF grants DMS-0652641, DMS-0652652, DMS-1001256, the “Research in Pairs” program by the Mathematiches Forschungsinstitut Oberwolfach in 2011, and the Hausdorff Institut in Bonn.

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Lenart, C., Schilling, A. Crystal energy functions via the charge in types A and C . Math. Z. 273, 401–426 (2013). https://doi.org/10.1007/s00209-012-1011-2

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