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

Boris Feigin ^[1] ; Igor Makhlin ^[2]
1. [1] Higher School of Economics, National Research University
  
  Higher School of Economics, National Research University
  
  Rusia
2. [2] Landau Institute for Theoretical Physics
  
  Landau Institute for Theoretical Physics
  
  Rusia
Localización: Selecta Mathematica, New Series, ISSN 1022-1824, Vol. 22, Nº. 3, 2016, págs. 1703-1747
Idioma: inglés
DOI: 10.1007/s00029-016-0223-4
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Resumen
- We present a new combinatorial formula for Hall–Littlewood functions associated with the affine root system of type A˜n−1, i.e., corresponding to the affine Lie algebra sln. 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.