论文标题
旋转Q-Whittaker多项式和变形量子TODA
Spin q-Whittaker polynomials and deformed quantum Toda
论文作者
论文摘要
Borodin and Wheeler（Arxiv：1701.06292）最近在可集成的$ \ Mathfrak {SL} _2 $ Vertex型号的背景下引入了Spin $ Q $ - 旋转对称的多项式$λ$标记的$λ$。它们是$ t = 0 $麦克唐纳多项式的单参数变形。我们提出了一种新的，更方便的旋转$ q $ - 惠塔克多项式的修改，并分别在这些多项式中与特征值分别在对角线上进行对角线作用的两个麦克唐纳德型$ q $ - 差异操作员，$ q^{ - λ_1} $和$ q^{ - q^{λ_n} $（$λ我们研究了基于自旋$ Q $ Whittaker多项式的交织阵列的概率度量，并将其可观察物与已知的随机粒子系统（例如$ Q $ -HAHN TASEP）匹配。 在缩放限制为$ q \附近1 $中，旋转$ q $ - 惠泰克多项式变成了$ \ mathfrak {gl} _n _n $ whittaker函数的新的单参数变形。恢复后的Pieri类型规则导致量子Toda Hamiltonian的单参数变形。变形的汉密尔顿人对我们的新旋转惠特克功能对角线起作用。在随机方面，作为$ q \近1 $，我们发现了Barraquand和Corwin的Beta聚合物模型的多级扩展（Arxiv：1503.04117），并将其与Spin Whittaker功能相关联。
Spin $q$-Whittaker symmetric polynomials labeled by partitions $λ$ were recently introduced by Borodin and Wheeler (arXiv:1701.06292) 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, $q^{-λ_1}$ and $q^{λ_N}$ (where $λ$ 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 (arXiv:1503.04117), and relate it to spin Whittaker functions.
