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Positive Level, Negative Level and Level Zero

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Schubert Calculus and Its Applications in Combinatorics and Representation Theory (ICTSC 2017)

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Abstract

This is a survey on the combinatorics and geometry of integrable representations of quantum affine algebras with a particular focus on level 0. Pictures and examples are included to illustrate the affine Weyl group orbits, crystal graphs and Macdonald polynomials that provide detailed understanding of the structure of the extremal weight modules and their characters. The final section surveys the alcove walk method of working with the positive level, negative level and level zero affine flag varieties and describes the corresponding actions of the affine Hecke algebra.

Dedicated to I. G. Macdonald and A. O. Morris.

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Acknowledgements

We thank Martha Yip for conversations, calculations and teaching us so much about Macdonald polynomials over the years. We thank all the institutions which have supported our work on this paper, particularly the University of Melbourne and the Australian Research Council (grants DP1201001942, DP130100674 and DE190101231). Arun Ram thanks IHP (Institute Henri Poincaré) and ICERM (Institute for Computational and Experimental Research in Mathematics) for support, hospitality and a stimulating working environments at the thematic programs “Automorphic Forms, Combinatorial Representation Theory and Multiple Dirichlet Series” and “Combinatorics and Interactions”.

Finally, it is a pleasure to dedicate this paper to Ian Macdonald and Alun Morris who forcefully led the way to the kinds explorations of affine combinatorial representation theory that are happening these days.

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  1. School of Mathematics and Statistics, University of Melbourne, Parkville, VIC, 3010, Australia

    Finn McGlade, Arun Ram & Yaping Yang

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  1. Finn McGlade
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Correspondence to Arun Ram .

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Editors and Affiliations

  1. School of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong, China

    Jianxun Hu

  2. School of Mathematics, Sun Yat-sen University, Guangzhou, Guangdong, China

    Changzheng Li

  3. Department of Mathematics, Virginia Tech, Blacksburg, VA, USA

    Leonardo C. Mihalcea

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McGlade, F., Ram, A., Yang, Y. (2020). Positive Level, Negative Level and Level Zero. In: Hu, J., Li, C., Mihalcea, L.C. (eds) Schubert Calculus and Its Applications in Combinatorics and Representation Theory. ICTSC 2017. Springer Proceedings in Mathematics & Statistics, vol 332. Springer, Singapore. https://doi.org/10.1007/978-981-15-7451-1_7

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