A bijective proof of Kohnert's rule for Schubert polynomials

A Unified Approach to Combinatorial Formulas for Schubert Polynomials

Schubert polynomials were introduced in the context of the geometry of flag varieties. This paper investigates some of the connections not yet understood between several combinatorial structures for the construction of Schubert polynomials; we also present simplifications in some of the existing approaches to this area. We designate certain line diagrams known as rc-graphs as the main structure. The other structures in the literature we study include: semistandard Young tableaux, Kohnert diagrams, and balanced labelings of the diagram of a permutation. The main tools in our investigation are c…

Demazure特征标与Schur多项式相乘的插入算法

An insertion algorithm for multiplying Demazure characters by Schur polynomials

本文在Kohnert组合模型上引入一种推广RSK插入的算法，给出Demazure特征标与Schur多项式乘积的显式非负公式，部分解决Polo关于Demazure模张量积具有Schubert滤过的猜想。

Coxeter combinatorics and spherical Schubert geometry

For a finite Coxeter system and a subset of its diagram nodes, we define spherical elements (a generalization of Coxeter elements). Conjecturally, for Weyl groups, spherical elements index Schubert varieties in a flag manifold G/B that are spherical for the action of a Levi subgroup. We evidence the conjecture, employing the combinatorics of Demazure modules, and work of R. Avdeev-A. Petukhov, M. Can-R. Hodges, R. Hodges-V. Lakshmibai, P. Karuppuchamy, P. Magyar-J. Weyman-A. Zelevinsky, N. Perrin, J. Stembridge, and B. Tenner. In type A, we establish connections with the key polynomials of A. …

A tableau formula for vexillary Schubert polynomials in type C

Ikeda-Mihalcea-Naruse's double Schubert polynomials represent the equivariant cohomology classes of Schubert varieties in the type C flag varieties. The goal of this paper is to obtain a new tableau formula of these polynomials associated to vexillary signed permutations introduced by Anderson-Fulton. To achieve that goal, we introduce flagged factorial (Schur) $Q$-functions, combinatorially defined in terms of (flagged) marked shifted tableaux for flagged strict partitions, and prove their Schur-Pfaffian formula. As an application, we also obtain a new combinatorial formula of factorial $Q$-f…

A generalization of Edelman--Greene insertion for Schubert polynomials

Edelman and Greene generalized the Robinson--Schensted--Knuth correspondence to reduced words in order to give a bijective proof of the Schur positivity of Stanley symmetric functions. Stanley symmetric functions may be regarded as the stable limits of Schubert polynomials, and similarly Schur functions may be regarded as the stable limits of Demazure characters for the general linear group. We modify the Edelman--Greene correspondence to give an analogous, explicit formula for the Demazure character expansion of Schubert polynomials. Our techniques utilize dual equivalence and its polynomial …

上下方案的混合管道梦

Hybrid pipe dreams for the lower-upper scheme

本文推广混合管道梦至通用情形，用基于杨-巴克斯特的方法证明等变下的混合独立性与递推关系，其多项式可计算上下簇的等变类，并给出两种证明，还提出通量定义以推导管道梦概念。

量子无碰撞管道梦想

Quantum bumpless pipe dreams

针对量子双舒伯特多项式缺乏组合表述的问题，提出量子无碰撞管道梦想的推广概念，给出其作为量子无碰撞管道梦想二项式权重之和的组合公式，并通过双射证明验证了公式正确性。

Doppelgängers: Bijections of Plane Partitions

We say two posets are "doppelgängers" if they have the same number of $P$-partitions of each height $k$. We give a uniform framework for bijective proofs that posets are doppelgängers by synthesizing $K$-theoretic Schubert calculus techniques of H. Thomas and A. Yong with M. Haiman's rectification bijection and an observation of R. Proctor. Geometrically, these bijections reflect the rational equivalence of certain subvarieties of minuscule flag manifolds. As a special case, we provide the first bijective proof of a 1983 theorem of R. Proctor---that plane partitions of height $k$ in a rectangl…

Transition formulas for involution Schubert polynomials

The orbits of the orthogonal and symplectic groups on the flag variety are in bijection, respectively, with the involutions and fixed-point-free involutions in the symmetric group $S_n$. Wyser and Yong have described polynomial representatives for the cohomology classes of the closures of these orbits, which we denote as $\hat{\mathfrak{S}}_y$ (to be called involution Schubert polynomials) and $\hat{\mathfrak{S}}^{\tt FPF}_y$ (to be called fixed-point-free involution Schubert polynomials). Our main results are explicit formulas decomposing the product of $\hat{\mathfrak{S}}_y$ (respectively, $…

Pattern bounds for principal specializations of $β$-Grothendieck Polynomials

There has been recent interest in lower bounds for the principal specializations of Schubert polynomials $ν_w := \mathfrak S_w(1,\dots,1)$. We prove a conjecture of Yibo Gao in the setting of $1243$-avoiding permutations that gives a lower bound for $ν_w$ in terms of the permutation patterns contained in $w$. We extended this result to principal specializations of $β$-Grothendieck polynomials $ν^{(β)}_w := \mathfrak G^{(β)}_w(1,\dots,1)$ by restricting to the class of vexillary $1243$-avoiding permutations. Our methods are bijective, offering a combinatorial interpretation of the coefficients …