An iterative-bijective approach to generalizations of Schur's theorem

Bijective Proofs of Shifted Tableau and Alternating Sign Matrix Identities

We give a bijective proof of an identity relating primed shifted gl(n)-standard tableaux to the product of a gl(n) character in the form of a Schur function and a product of sums of x and y terms. This result generalises a number of well--known results due to Robbins and Rumsey, Chapman, Tokuyama, Okada and Macdonald. An analogous result is then obtained in the case of primed shifted sp(2n)-standard tableaux which are bijectively related to the product of a t-deformed sp(2n) character and another x and y product. All results are also interpreted in terms of alternating sign matrix identities, …

匹配中短弦的Schur正性

Schur-Positivity of Short Chords in Matchings

本文研究了固定未匹配顶点数的匹配集合在短弦下的Schur正性，通过两种方法证明了这一性质，并探讨了Schur展开系数与贝塞尔多项式的关系，以及匹配的等价关系与不可约表示的联系，最终刻画了所有满足特定条件的匹配。

Skew quantum Murnaghan-Nakayama rule

In this paper, we extend recent results of Assaf and McNamara on skew Pieri rule and skew Murnaghan-Nakayama rule to a more general identity, which gives an elegant expansion of the product of a skew Schur function with a quantum power sum function in terms of skew Schur functions. We give two proofs, one completely bijective in the spirit of Assaf-McNamara's original proof, and one via Lam-Lauve-Sotille's skew Littlewood-Richardson rule. We end with some conjectures for skew rules for Hall-Littlewood polynomials.

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 …

斜 Schur 模的非迭代拉直算法与正交性

A non-iterative straightening algorithm and orthogonality for skew Schur modules

本文将 Fulton 的 Schur 模行列式构造推广至斜设置，提出了一种非迭代拉直算法以显式闭系数公式将任意元素表示为新的 D-基，并证明该基源于对半标准表基的 Gram-Schmidt 正交化，从而揭示了自然内积并将拉直重表述为正交投影。

An extension of MacMahon's Equidistribution Theorem to ordered multiset partitions

A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work we prove a strengthening of this theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof is bijective and involves a new generalization of Carlitz's insertion method. This generalization leads to a new extension of Macdonald polynomials for hook shapes. We use our main theorem to show that these p…

拉姆齐理论中的指数舒尔定理与指数欣德曼定理

Exponential Schur and Hindman Theorem in Ramsey Theory

本文给出指数舒尔定理的两种简短证明，证明指数欣德曼定理，得出相关分拆正则方程及单色序列推论，还证明特定超滤不存在，推进指数拉姆齐理论研究。

Spin Kostka 多项式与顶点算子

Spin Kostka polynomials and vertex operators

本文通过顶点算子给出 spin Kostka-Foulkes 多项式的迭代公式，获得平行性质与标记表计数公式，并证实 Aokage 猜想。