A basis for the right quantum algebra and the "1=q" principle

对称群群代数的左到右极小值基（更新版本）

The left-to-right minima basis of the group algebra of the symmetric group (updated version)

本文提出了一种基于排列左到右极小值集的新基，该基在下降代数作用下呈现三角性质，为对称群群代数提供了一种新的结构分析方法。

Halfway to Rota's basis conjecture

In 1989, Rota made the following conjecture. Given $n$ bases $B_{1},\dots,B_{n}$ in an $n$-dimensional vector space $V$, one can always find $n$ disjoint bases of $V$, each containing exactly one element from each $B_{i}$ (we call such bases transversal bases). Rota's basis conjecture remains wide open despite its apparent simplicity and the efforts of many researchers (for example, the conjecture was recently the subject of the collaborative "Polymath" project). In this paper we prove that one can always find $\left(1/2-o\left(1\right)\right)n$ disjoint transversal bases, improving on the pre…

可脚排序袜子排列的基础

The Basis of Foot-Sortable Sock Orderings

本文研究了可脚排序袜子排列的基础，明确描述了无法通过堆叠和弹出操作排序的最小初始袜子排列。

A型Peterson Schubert演算的几何与左右图

Geometry of Peterson Schubert calculus in type A and left-right diagrams

本文通过构造Peterson簇上同调环的几何加法基，从几何角度解释结构常数正性并给出正组合公式，同时证明该基与Harada-Tymoczko基一致。

直角 Artin 群与上同调基图

Right-angled Artin groups and the cohomology basis graph

本文通过直角 Artin 群上同调中的杯积构造上同调基图，证明其总包含定义图作为子图，从而可从群的上同调重构原图并恢复平面性等图论不变量，同时探讨了该构造在初等子 minor 下的行为。

Cellularity of the p-Canonical Basis for Symmetric Groups

For symmetric groups we show that the p-canonical basis can be extended to a cell datum for the Iwahori-Hecke algebra H and that the two-sided p-cell preorder coincides with the Kazhdan-Lusztig two-sided cell preorder. Moreover, we show that left (or right) p-cells inside the same two-sided p-cell for Hecke algebras of finite crystallographic Coxeter systems are incomparable (Property A).