Frozen pipes: lattice models for Grothendieck polynomials
Frozen pipes: lattice models for Grothendieck polynomials
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DOI：
10.5802/alco.277
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发表时间：
2020-07
影响因子：
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通讯作者：
Ben Brubaker;Claire Fréchette;A. Hardt;Emily Tibor;Katherine Weber
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文献类型：
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作者：
Ben Brubaker;Claire Fréchette;A. Hardt;Emily Tibor;Katherine Weber
我们证明了几个不同的可解晶格模型族的存在性，它们的配分函数给出了任意排列的二重$\beta$-Grothendieck多项式和对偶双$\beta$-Grothendieck多项式。此外，我们引入了一类新的依赖于一对置换的双“双轴”-Grothendieck多项式，它同时推广了双多项式和对偶双多项式。然后利用这些模型和它们的杨-巴克斯特方程，对$\ β $-Grothendieck多项式的cochy恒等式进行了修正，将其推广到双轴$\ β $-Grothendieck多项式的新的柯西恒等式，并证明了二重$\ β $-Grothendieck多项式的一个新的分支规则。
We prove the existence of several different families of solvable lattice models whose partition functions give the double $\beta$-Grothendieck polynomials and the dual double $\beta$-Grothendieck polynomials for arbitrary permutations. Moreover, we introduce a new family of double "biaxial" $\beta$-Grothendieck polynomials depending on a pair of permutations which simultaneously generalize both the double and dual double polynomials. We then use these models and their Yang-Baxter equations to reprove Fomin-Kirillov's Cauchy identity for $\beta$-Grothendieck polynomials, generalize it to a new Cauchy identity for biaxial $\beta$-Grothendieck polynomials, and prove a new branching rule for double $\beta$-Grothendieck polynomials.