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A combinatorial formula for affine Hall–Littlewood functions via a weighted Brion theorem

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Abstract

We present a new combinatorial formula for Hall–Littlewood functions associated with the affine root system of type \({{\tilde{A}}}_{n-1}\), i.e., corresponding to the affine Lie algebra \({{\widehat{\mathfrak {sl}}}}_n\). Our formula has the form of a sum over the elements of a basis constructed by Feigin, Jimbo, Loktev, Miwa and Mukhin in the corresponding irreducible representation. Our formula can be viewed as a weighted sum of exponentials of integer points in a certain infinite-dimensional convex polyhedron. We derive a weighted version of Brion’s theorem and then apply it to our polyhedron to prove the formula.

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Authors and Affiliations

  1. International Laboratory of Representation Theory and Mathematical Physics, National Research University Higher School of Economics, Myasnitskaya ulitsa 20, Moscow, Russia, 101000

    Boris Feigin & Igor Makhlin

  2. Landau Institute for Theoretical Physics, prospekt Akademika Semenova 1A, Chernogolovka, Russia, 142432

    Boris Feigin & Igor Makhlin

Authors
  1. Boris Feigin
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  2. Igor Makhlin
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Correspondence to Igor Makhlin.

Additional information

Igor Makhlin: Supported in part by the Simons Foundation and the Moebius Contest Foundation for Young Scientists.

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Feigin, B., Makhlin, I. A combinatorial formula for affine Hall–Littlewood functions via a weighted Brion theorem. Sel. Math. New Ser. 22, 1703–1747 (2016). https://doi.org/10.1007/s00029-016-0223-4

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  • DOI: https://doi.org/10.1007/s00029-016-0223-4

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