Selecta Mathematica

, Volume 24, Issue 2, pp 751–874 | Cite as

Higher spin six vertex model and symmetric rational functions

  • Alexei Borodin
  • Leonid PetrovEmail author


We consider a fully inhomogeneous stochastic higher spin six vertex model in a quadrant. For this model we derive concise integral representations for multi-point q-moments of the height function and for the q-correlation functions. At least in the case of the step initial condition, our formulas degenerate in appropriate limits to many known formulas of such type for integrable probabilistic systems in the (1\(+\)1)d KPZ universality class, including the stochastic six vertex model, ASEP, various q-TASEPs, and associated zero range processes. Our arguments are largely based on properties of a family of symmetric rational functions that can be defined as partition functions of the inhomogeneous higher spin six vertex model for suitable domains. In the homogeneous case, such functions were previously studied in Borodin (On a family of symmetric rational functions, 2014); they also generalize classical Hall–Littlewood and Schur polynomials. A key role is played by Cauchy-like summation identities for these functions, which are obtained as a direct corollary of the Yang–Baxter equation for the higher spin six vertex model.

Mathematics Subject Classification

Primary 60K35 Secondary 05E05 82B23 


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© Springer International Publishing 2016

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.Institute for Information Transmission ProblemsMoscowRussia
  3. 3.Department of MathematicsUniversity of VirginiaCharlottesvilleUSA

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