精细典范稳定格罗滕迪克多项式及其对偶，第二部分

Refined dual Grothendieck polynomials, integrability, and the Schur measure

We construct a vertex model whose partition function is a refined dual Grothendieck polynomial, where the states are interpreted as nonintersecting lattice paths. Using this, we show refined dual Grothendieck polynomials are multi-Schur functions and give a number of identities, including a Littlewood and Cauchy(-Littlewood) identity. We then refine Yeliussizov's connection between dual Grothendieck polynomials and the last passage percolation (LPP) stochastic process discussed by Johansson. By refining algebraic techniques of Johansson, we show Jacobi-Trudi formulas for skew refined dual Grot…

Vertex models for Canonical Grothendieck polynomials and their duals

We study solvable lattice models associated to canonical Grothendieck polynomials and their duals. We derive inversion relations and Cauchy identities.

关于典范稳定 Grothendieck 函数的一个注记

A note on canonical stable Grothendieck functions

本文提出了一种新方法证明稳定 Grothendieck 多项式的 Murnaghan-Nakayama 型法则，并为典范稳定 Grothendieck 函数建立了该法则。

自由费米子与典范格罗滕迪克多项式

Free-fermions and canonical Grothendieck polynomials

本文用自由费米子表示精细典范格罗滕迪克多项式及其斜版本，并由此推导出斜柯西恒等式、分支规则、展开公式与积分公式等多项恒等式。

Duality and deformations of stable Grothendieck polynomials

Stable Grothendieck polynomials can be viewed as a K-theory analog of Schur polynomials. We extend stable Grothendieck polynomials to a two-parameter version, which we call canonical stable Grothendieck functions. These functions have the same structure constants (with scaling) as stable Grothendieck polynomials, and (composing with parameter switching) are self-dual under the standard involutive ring automorphism. We study various properties of these functions, including combinatorial formulas, Schur expansions, Jacobi-Trudi type identities, and associated Fomin-Greene operators.

Crystal structures for canonical Grothendieck functions

We give a $U_q(\mathfrak{sl}_n)$-crystal structure on multiset-valued tableaux, hook-valued tableaux, and valued-set tableaux, whose generating functions are the weak symmetric, canonical, and dual weak symmetric Grothendieck functions, respectively. We show the result is isomorphic to a (generally infinite) direct sum of highest weight crystals, and for multiset-valued tableaux and valued-set tableaux, we provide an explicit bijection. As a consequence, these generating functions are Schur positive; in particular, the canonical Grothendieck functions, which was not previously known. We also g…

钩值表的去拥挤算法

Uncrowding algorithm for hook-valued tableaux

本文提出一种针对钩值表的新颖去拥挤算法，通过处理钩臂或钩腿条目生成集合值表与列标记递增表，并证明其与晶体算子交织，应用于典范格罗滕迪克多项式的展开。

钩值表格去拥挤与表格切换

Hook-valued tableau uncrowding and tableau switching

本文通过去拥挤和jeu de taquin算法，发现钩值表格与精美表格之间的联系，并基于表格切换的结果，提出双旗帜表格作为精细稳定Grothendieck多项式的新模型。

自由费米子与多Schur函数的Schur展开

Free fermions and Schur expansions of multi-Schur functions

本研究提出多Schur函数的自由费米子新表示，构造对应索引的费米子Fock空间基并验证玻色-费米对应关系，给出其在精细化对偶Grothendieck多项式下的展开方法及Hall对偶属于对称函数完备环的充分条件。

有限偏序集上的对偶复形与典范复形

Dualizing and canonical complexes on finite posets

本文发展了有限偏序集上的对偶复形理论及科恩-麦考利性理论，为相关代数几何研究奠定基础。

用 Pipe Dream 组合学进行基变换

Changing Bases with Pipe Dream Combinatorics

本文利用无碰撞管道梦 BPD 建立 Schubert 与 Grothendieck 多项式互展的显式组合规则，并推广到后稳定情形。

Tamari区间的小型乘积公式

Refined product formulas for Tamari intervals

本文提出用于计算Tamari格子标准复形和associahedra单元对角线的f向量的简短乘积公式，揭示了组合结构的数学特性。

混合 Grothendieck 多项式

Hybrid Grothendieck polynomials

本文定义混合 Grothendieck 多项式作为集值反向平面分割的权重生成函数，证明其对称性，通过晶体结构得到 Schur 展开，并推导饱和 Newton 多面体及 omega 对合下的组合公式。