论文标题
冷冻管道：Grothendieck多项式的晶格模型
Frozen Pipes: Lattice Models for Grothendieck Polynomials
论文作者
论文摘要
我们介绍了由两组分区$ v，w $ - 双轴双$（β，q）$ - Grothendieck多项式索引的两参数多元多项式的家庭 - 专门针对$ q = 0 $和$ v = 1 $ to double $β$β$β$ -GROTHENDIECK POTROTHENDIECK POTROSECK POTRUSECK-GROTHENDIECK POTORUS-equivAriant knectivic knectivic knectivic knectivic knectivic knectim knectim knectim incounctive knewore。最初是通过分隔差的操作员递归定义的，我们的主要结果是这些新的多项式作为可解决的晶格模型的分区函数出现。此外，在$ n $ n $变量中的多项式可解的模型的相关量子组是$ u_q（\ wideHat {\ mathfrak {sl}} _ {n+1}）$ u_q（\ wideHat {\ wideHat {\ wideHat {\ wideHat {\ wideHat {\ mathfrak {\ mathfrak {n+1}）$ $ r $ -mmatrix。通过利用晶格模型的产生的杨 - 巴克斯特方程，我们表明这些多项式同时概括了双$β$ - grothendieck多项式和双$β$ grothendieck tolynomials for interrudations for interrudations。然后，我们使用模型和扬 - 巴克斯特方程的属性来谴责Fomin-Kirillov的Cauchy身份，以$β$ - grothendieck多项式为多项式，将其推广到新的cauchy身份，用于双轴双$β$β$β$ - grophendieck polynomials，并支持新的分支$β$-β$ -GRNONPOTORNENONPOTORNENPOTORNECECK。
We introduce families of two-parameter multivariate polynomials indexed by pairs of partitions $v,w$ -- biaxial double $(β,q)$-Grothendieck polynomials -- which specialize at $q=0$ and $v=1$ to double $β$-Grothendieck polynomials from torus-equivariant connective K-theory. Initially defined recursively via divided difference operators, our main result is that these new polynomials arise as partition functions of solvable lattice models. Moreover, the associated quantum group of the solvable model for polynomials in $n$ pairs of variables is a Drinfeld twist of the $U_q(\widehat{\mathfrak{sl}}_{n+1})$ $R$-matrix. By leveraging the resulting Yang-Baxter equations of the lattice model, we show that these polynomials simultaneously generalize double $β$-Grothendieck polynomials and dual double $β$-Grothendieck polynomials for arbitrary permutations. We then use properties of the model and Yang-Baxter equations to reprove Fomin-Kirillov's Cauchy identity for $β$-Grothendieck polynomials, generalize it to a new Cauchy identity for biaxial double $β$-Grothendieck polynomials, and prove a new branching rule for double $β$-Grothendieck polynomials.
