Vertex models, TASEP and Grothendieck polynomials
Vertex models, TASEP and Grothendieck polynomials
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顶点模型、TASEP 和 Grothendieck 多项式
DOI：
10.1088/1751-8113/46/35/355201
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发表时间：
2013
期刊：
影响因子：
--
通讯作者：
K. Motegi and K. Sakai
中科院分区：
文献类型：
--
作者：
上島裕司;与儀千尋;辻孝祐;小島一男;Tomokatsu Hayakawa;K. Motegi and K. Sakai;Kohei Motegi;K. Motegi and K. Sakai
我们研究了一个单参数可积五顶点模型族的波函数及其标量积。在参数的一个特殊点上，所研究的模型与一个不可逆的相互作用随机粒子系统--所谓的完全不对称简单排斥过程（TASEP）有关。通过将量子逆散射方法与波函数的矩阵乘积表示相结合，将五顶点模型的开壳/离壳波函数表示为某种行列式形式.在一定的归一化因子下，我们发现波函数由Grothendieck多项式给出，它是Schur多项式的单参数变形。引入Grothendieck多项式的对偶形式，利用波函数标量积的行列式表示，导出了Grothendieck多项式及其导数满足的广义Cauchy恒等式.给出了Grothendieck多项式的几个表示理论公式。作为一个副产品，如绿色功能的周期TASEP的松弛动力学被发现描述的格罗滕迪克多项式。
We examine the wavefunctions and their scalar products of a one-parameter family of integrable five-vertex models. At a special point of the parameter, the model investigated is related to an irreversible interacting stochastic particle system—the so-called totally asymmetric simple exclusion process (TASEP). By combining the quantum inverse scattering method with a matrix product representation of the wavefunctions, the on-/off-shell wavefunctions of the five-vertex models are represented as a certain determinant form. Up to some normalization factors, we find that the wavefunctions are given by Grothendieck polynomials, which are a one-parameter deformation of Schur polynomials. Introducing a dual version of the Grothendieck polynomials, and utilizing the determinant representation for the scalar products of the wavefunctions, we derive a generalized Cauchy identity satisfied by the Grothendieck polynomials and their duals. Several representation theoretical formulae for the Grothendieck polynomials are also presented. As a byproduct, the relaxation dynamics such as Green functions for the periodic TASEP are found to be described in terms of the Grothendieck polynomials.