Determinantal formulas for dual Grothendieck polynomials

Jacobi--Trudi formulas for flagged refined dual stable Grothendieck polynomials

Recently Galashin, Grinberg, and Liu introduced the refined dual stable Grothendieck polynomials, which are symmetric functions in $x=(x_1,x_2,\dots)$ with additional parameters $t=(t_1,t_2,\dots)$. The refined dual stable Grothendieck polynomials are defined as a generating function for reverse plane partitions of a given shape. They interpolate between Schur functions and dual stable Grothendieck polynomials introduced by Lam and Pylyavskyy in 2007. Flagged refined dual stable Grothendieck polynomials are a more refined version of refined dual stable Grothendieck polynomials, where lower and…

Symmetric Grothendieck polynomials, skew Cauchy identities, and dual filtered Young graphs

Symmetric Grothendieck polynomials are analogues of Schur polynomials in the K-theory of Grassmannians. We build dual families of symmetric Grothendieck polynomials using Schur operators. With this approach we prove skew Cauchy identity and then derive various applications: skew Pieri rules, dual filtrations of Young's lattice, generating series and enumerative identities. We also give a new explanation of the finite expansion property for products of Grothendieck polynomials.

Free-fermions and skew stable Grothendieck polynomials

Skew stable Grothendieck polynomials are $K$-theoretic analogues of skew Schur polynomials. We give a free-fermionic presentation of skew stable Grothendieck polynomials and their dual symmetric functions. By using our presentation, we derive a family of determinantal formulas, which are $K$-analogues of the Jacobi-Trudi formula for skew Schur functions. We also introduce a combinatorial method to calculate certain expansions of skew (dual) stable Grothendieck polynomials by using the non-commutative supersymmetric Schur functions.

精化典范稳定格罗滕迪克多项式及其对偶，第一部分

Refined canonical stable Grothendieck polynomials and their duals, Part 1

本文提出带两列无穷参数的精细典范稳定格罗滕迪克多项式及其对偶，统一了多个已有推广，并建立了Jacobi–Trudi型公式、组合模型、Schur展开、Schur正性与对偶性。

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.

On the Pieri rules of stable and dual stable Grothendieck polynomials

We give an explanation for the Pieri coefficients for the stable and dual stable Grothendieck polynomials; their non-leading terms are obtained by taking an alternating sum of meets (or joins) of their leading terms.

A Littlewood-Richardson Rule for Dual Stable Grothendieck Polynomials

For a given skew shape, we build a crystal graph on the set of all reverse plane partitions that have this shape. As a consequence, we get a simple extension of the Littlewood-Richardson rule for the expansion of the corresponding dual stable Grothendieck polynomial in terms of Schur polynomials.

The Stembridge Equality for Skew Stable Grothendieck Polynomials and Skew Dual Stable Grothendieck Polynomials

The Schur polynomials $s_λ$ are essential in understanding the representation theory of the general linear group. They also describe the cohomology ring of the Grassmannians. For $ρ= (n, n-1, \dots, 1)$ a staircase shape and $μ\subseteq ρ$ a subpartition, the Stembridge equality states that $s_{ρ/μ} = s_{ρ/μ^T}$. This equality provides information about the symmetry of the cohomology ring. The stable Grothendieck polynomials $G_λ$, and the dual stable Grothendieck polynomials $g_λ$, developed by Buch, Lam, and Pylyavskyy, are variants of the Schur polynomials and describe the $K$-theory of the…

Coincidences among skew dual stable Grothendieck polynomials

The question of when two skew Young diagrams produce the same skew Schur function has been well-studied. We investigate the same question in the case of stable Grothendieck polynomials, which are the K-theoretic analogues of the Schur functions. We prove a necessary condition for two skew shapes to give rise to the same dual stable Grothendieck polynomial. We also provide a necessary and sufficient condition in the case where the two skew shapes are ribbons.

斜对称矩阵Schubert簇的Castelnuovo-Mumford正则度

Castelnuovo-Mumford regularity of skew-symmetric matrix Schubert varieties

本文通过给出辛Grothendieck多项式次数的组合公式，计算了斜对称矩阵Schubert簇的Castelnuovo-Mumford正则度，并刻画了其最高次齐次分量及最大正则度。

Degrees of symmetric Grothendieck polynomials and Castelnuovo-Mumford regularity

We give an explicit formula for the degree of the Grothendieck polynomial of a Grassmannian permutation and a closely related formula for the Castelnuovo-Mumford regularity of the Schubert determinantal ideal of a Grassmannian permutation. We then provide a counterexample to a conjecture of Kummini-Lakshmibai-Sastry-Seshadri on a formula for regularities of standard open patches of particular Grassmannian Schubert varieties and show that our work gives rise to an alternate explicit formula in these cases. We end with a new conjecture on the regularities of standard open patches of arbitrary Gr…

精细典范稳定格罗滕迪克多项式及其对偶，第二部分

Refined canonical stable Grothendieck polynomials and their duals, Part 2

本文为精细典范稳定格罗滕迪克多项式及其对偶的第二部分，利用集值杨表与反向平面分割的推广给出组合解释，并推广至带旗与偏斜版本。

Automorphisms on the ring of symmetric functions and stable and dual stable Grothendieck polynomials

The dual stable Grothendieck polynomials $g_λ$ and their sums $\sum_{μ\subsetλ} g_μ$ (which represent $K$-homology classes of boundary ideal sheaves and structure sheaves of Schubert varieties in the Grassmannians) have the same product structure constants. In this paper we first explain that the ring automorphism $g_λ\mapsto\sum_{μ\subsetλ} g_μ$ on the ring of symmetric functions is described as the operator $F^\perp$, the adjoint of the multiplication $(F\cdot)$, by a "group-like" element $F=\sum_{i} h_i$ where $h_i$ is the complete symmetric function. Next we give a generalization: starting…

Subword complexes via triangulations of root polytopes

Subword complexes are simplicial complexes introduced by Knutson and Miller to illustrate the combinatorics of Schubert polynomials and determinantal ideals. They proved that any subword complex is homeomorphic to a ball or a sphere and asked about their geometric realizations. We show that a family of subword complexes can be realized geometrically via regular triangulations of root polytopes. This implies that a family of $β$-Grothendieck polynomials are special cases of reduced forms in the subdivision algebra of root polytopes. We can also write the volume and Ehrhart series of root polyto…

Grothendieck 多项式 tableau 模型的统一框架

Unified framework for tableau models of Grothendieck polynomials

本文构建统一组合框架，涵盖多种 tableau 模型，给出 Grothendieck 多项式两类对偶性的组合证明，并首次将第二类对偶性推广到精化情形。

交错游程多项式的行列式表示

Determinantal representations of alternating run polynomials

本文基于函数商高阶导数的行列式公式，推导了Eulerian多项式、Andre多项式及多种交错游程多项式的行列式表示，并揭示了其递推关系及与对偶Stirling排列的联系。

Dual Grothendieck polynomials via last-passage percolation

The ring of symmetric functions has a basis of dual Grothendieck polynomials that are inhomogeneous $K$-theoretic deformations of Schur polynomials. We prove that dual Grothendieck polynomials determine column distributions for a directed last-passage percolation model.

Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions

The dual stable Grothendieck polynomials are a deformation of the Schur functions, originating in the study of the K-theory of the Grassmannian. We generalize these polynomials by introducing a countable family of additional parameters, and we prove that this generalization still defines symmetric functions. For this fact, we give two self-contained proofs, one of which constructs a family of involutions on the set of reverse plane partitions generalizing the Bender-Knuth involutions on semistandard tableaux, whereas the other classifies the structure of reverse plane partitions with entries 1…

Grothendieck多项式的Cauchy恒等式与通过pipe dreams的对偶RSK对应

Cauchy identities for Grothendieck polynomials and a dual RSK correspondence through pipe dreams

本文研究双重Grothendieck多项式的Cauchy恒等式，提出pipe dream矫正算法构建pipe dreams间的权重保持双射，并基于新流算子揭示其对称性，最终恢复对偶RSK对应变体。

Notes on Schubert, Grothendieck and Key Polynomials

We introduce common generalization of (double) Schubert, Grothendieck, Demazure, dual and stable Grothendieck polynomials, and Di Francesco-Zinn-Justin polynomials. Our approach is based on the study of algebraic and combinatorial properties of the reduced rectangular plactic algebra and associated Cauchy kernels.

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.

关于混合格罗滕迪克多项式的一个拉姆 - 波斯尼科夫 - 皮利亚夫斯基不等式

A Lam--Postnikov--Pylyavskyy inequality for hybrid Grothendieck polynomials

证明了混合格罗滕迪克多项式的多元拉姆 - 波斯尼科夫 - 皮利亚夫斯基型不等式，统一并细化了相关结果，还猜想了该不等式及托马斯 - 杨猜想在（等变）舒伯特和格罗滕迪克多项式情形的扩展。

通过泡化（bubbling）证明Grothendieck多项式的M-凸性

M-convexity of Grothendieck polynomials via bubbling

该研究引入泡化图计算vexillary排列的Grothendieck多项式支撑集，证明其最高齐次分量支撑集与显式带旗Weyl模对偶特征标支撑集一致，且齐次化Grothendieck多项式具有M-凸支撑集。

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…

Structure constants for K-theory of Grassmannians, revisited

The problem of computing products of Schubert classes in the cohomology ring can be formulated as the problem of expanding skew Schur polynomials into the basis of ordinary Schur polynomials. In contrast, the problem of computing the structure constants of the Grothendieck ring of a Grassmannian variety with respect to its basis of Schubert structure sheaves is not equivalent to expanding skew stable Grothendieck polynomials into the basis of ordinary stable Grothendiecks. Instead, we show that the appropriate K-theoretic analogy is through the expansion of skew reverse plane partitions into t…

带标志的精炼斜稳定格罗滕迪克多项式的密钥扩展

Key expansion of the flagged refined skew stable Grothendieck polynomial

本文提出了一种将带标志的精炼斜稳定格罗滕迪克多项式用密钥多项式展开的组合方法，并扩展了其他多项式在稳定格罗滕迪克多项式和对偶稳定格罗滕迪克多项式下的展开形式。

A bijective proof of the Cauchy identity for Grothendieck polynomials

We consider pairs of a set-valued column-strict tableau and a reverse plane partition of the same shape. We introduce algortithms for them, which implies a bijective proof for the finite sum Cauchy identity for Grothendieck polynomials and dual Grothendieck polynomials.

Algebraic Structures on Grothendieck Groups of a Tower of Algebras

The Grothendieck group of the tower of symmetric group algebras has a self-dual graded Hopf algebra structure. Inspired by this, we introduce by way of axioms, a general notion of a tower of algebras and study two Grothendieck groups on this tower linked by a natural paring. Using representation theory, we show that our axioms give a structure of graded Hopf algebras on each Grothendieck groups and these structures are dual to each other. We give some examples to indicate why these axioms are necessary. We also give auxiliary results that are helpful to verify the axioms. We conclude with some…

Combinatorics of the K-theory of affine Grassmannians

We introduce a family of tableaux that simultaneously generalizes the tableaux used to characterize Grothendieck polynomials and k-Schur functions. We prove that the polynomials drawn from these tableaux are the affine Grothendieck polynomials and k-K-Schur functions -- Schubert representatives for the K-theory of affine Grassmannians and their dual in the nil Hecke ring. We prove a number of combinatorial properties including Pieri rules.

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。

Frozen Pipes: Lattice Models for Grothendieck Polynomials

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}}_…

移位对偶稳定 Grothendieck 多项式的组合公式

Combinatorial formulas for shifted dual stable Grothendieck polynomials

本文证明了关于偶 $K$理论 Schur $P$-和 $Q$-函数权生成函数的猜想，推导了其在 $ω$ 对合下的生成函数公式，并验证了 $GQ$-函数构成环基的猜想。

Integrability approach to Feher-Nemethi-Rimanyi-Guo-Sun type identities for factorial Grothendieck polynomials

Recently, Guo and Sun derived an identity for factorial Grothendieck polynomials which is a generalization of the one for Schur polynomials by Fehér, Némethi and Rimányi. We analyze the identity from the point of view of quantum integrability, based on the correspondence between the wavefunctions of a five-vertex model and the Grothendieck polynomials. We give another proof using the quantum inverse scattering method. We also apply the same idea and technique to derive an identity for factorial Grothendieck polynomials for rectangular Young diagrams. Combining with the Guo-Sun identity, we get…

Littlewood-Richardson coefficients for Grothendieck polynomials from integrability

We study the Littlewood-Richardson coefficients of double Grothendieck polynomials indexed by Grassmannian permutations. Geometrically, these are the structure constants of the equivariant $K$-theory ring of Grassmannians. Representing the double Grothendieck polynomials as partition functions of an integrable vertex model, we use its Yang-Baxter equation to derive a series of product rules for the former polynomials and their duals. The Littlewood-Richardson coefficients that arise can all be expressed in terms of puzzles without gashes, which generalize previous puzzles obtained by Knutson-T…

行列式簇K多项式的组合推导研究

Towards combinatorial derivations of K-polynomials for determinantal varieties

本文针对行列式簇的K多项式组合推导问题，提出对秩不超过k的复矩阵簇希尔伯特级数相关公式给出直接组合证明的目标，在k=1或k=min{m,n}-1时通过Littlewood–Richardson表征集上的显式符号反转对合完成了证明。

Kirillov 关于 Hecke-Grothendieck 多项式的猜想

Kirillov's conjecture on Hecke-Grothendieck polynomials

本文利用统计力学中的代数方法，将多参数多项式表示为新一类可解格子模型的配分函数，证明了 Kirillov 关于 Hecke-Grothendieck 多项式正性猜想，并展示了更大类别的多项式中罕见的负系数实例。

A Gröbner basis for Kazhdan-Lusztig ideals

Kazhdan-Lusztig ideals, a family of generalized determinantal ideals investigated in [Woo-Yong '08], provide an explicit choice of coordinates and equations encoding a neighbourhood of a torus-fixed point of a Schubert variety on a type A flag variety. Our main result is a Grobner basis for these ideals. This provides a single geometric setting to transparently explain the naturality of pipe dreams on the Rothe diagram of a permutation, and their appearance in: * combinatorial formulas [Fomin-Kirillov '94] for Schubert and Grothendieck polynomials of [Lascoux-Schutzenberger '82]; * the equivar…