Spin q-Whittaker Polynomials and Deformed Quantum Toda

Abstract

Spin q-Whittaker symmetric polynomials labeled by partitions \(\lambda \) were recently introduced by Borodin and Wheeler (Spin q-Whittaker Polynomials, 2017. arXiv preprint arXiv:1701.06292 [math.CO]) in the context of integrable \(\mathfrak {sl}_2\) vertex models. They are a one-parameter deformation of the \(t=0\) Macdonald polynomials. We present a new more convenient modification of spin q-Whittaker polynomials and find two Macdonald type q-difference operators acting diagonally in these polynomials with eigenvalues, respectively, \(q^{-\lambda _1}\) and \(q^{\lambda _N}\) (where \(\lambda \) is the polynomial’s label). We study probability measures on interlacing arrays based on spin q-Whittaker polynomials, and match their observables with known stochastic particle systems such as the q-Hahn TASEP. In a scaling limit as \(q\nearrow 1\), spin q-Whittaker polynomials turn into a new one-parameter deformation of the \(\mathfrak {gl}_n\) Whittaker functions. The rescaled Pieri type rule gives rise to a one-parameter deformation of the quantum Toda Hamiltonian. The deformed Hamiltonian acts diagonally on our new spin Whittaker functions. On the stochastic side, as \(q\nearrow 1\) we discover a multilevel extension of the beta polymer model of Barraquand and Corwin (Probab Theory Relat Fields 167(3–4):1057–1116, 2016. arXiv:1503.04117 [math.PR]), and relate it to spin Whittaker functions.