Minimal and maximal elements in Kazhdan-Lusztig double sided cells of $S_n$ and Robinson-Schensted correspondance

仿射A型Kazhdan-Lusztig胞腔中的单值性

Monodromy in Kazhdan-Lusztig cells in affine type A

本文利用仿射Robinson-Schensted对应描述仿射A型Kazhdan-Lusztig胞腔中双向边结构，等价于全面刻画仿射置换的Knuth等价类。

Matrix-Ball Construction of affine Robinson-Schensted correspondence

In his study of Kazhdan-Lusztig cells in affine type $A$, Shi has introduced an affine analog of Robinson-Schensted correspondence. We generalize the Matrix-Ball Construction of Viennot and Fulton to give a more combinatorial realization of Shi's algorithm. As a biproduct, we also give a way to realize the affine correspondence via the usual Robinson-Schensted bumping algorithm. Next, inspired by Lusztig and Xi, we extend the algorithm to a bijection between extended affine symmetric group and triples $(P, Q, ρ)$ where $P$ and $Q$ are tabloids and $ρ$ is a dominant weight. The weights $ρ$ get …

The Robinson-Schensted Correspondence and $A_2$-web Bases

We study natural bases for two constructions of the irreducible representation of the symmetric group corresponding to $[n,n,n]$: the {\em reduced web} basis associated to Kuperberg's combinatorial description of the spider category; and the {\em left cell basis} for the left cell construction of Kazhdan and Lusztig. In the case of $[n,n]$, the spider category is the Temperley-Lieb category; reduced webs correspond to planar matchings, which are equivalent to left cell bases. This paper compares the images of these bases under classical maps: the {\em Robinson-Schensted algorithm} between perm…

Sign insertion and Kazhdan-Lusztig cells of affine symmetric groups

Combinatorics of Kazhdan-Lusztig cells in affine type $A$ was originally developed by Lusztig, Shi, and Xi. Building on their work, Chmutov, Pylyavskyy, and Yudovina introduced the affine matrix-ball construction (abbreviated AMBC) which gives an analog of Robinson-Schensted correspondence for affine symmetric groups. An alternative approach to Kazhdan-Lusztig theory in affine type $A$ was developed by Blasiak in his work on catabolism. He introduced sign insertion algorithm and conjectured that if one fixes the two-sided cell, the recording tableau of sign insertion process determines uniquel…

双仿射罗宾逊-斯文斯坦对应

Dual Affine Robinson-Schensted Correspondence

本文提出了双仿射罗宾逊-斯文斯坦对应，通过构造一个双射，将扩展仿射对称群与元组(\bar{P},\bar{Q},λ,N)联系起来，其中\bar{P}和\bar{Q}是表格，λ是一个分割，N是一个整数，且需满足兼容条件。该构造推广了福明的生长图和维诺托的影子线，用于经典罗宾逊-斯文斯坦对应，且与仿射矩阵球构造以及施的对应是双的，因为P-表格相同，而Q-表格通过仿射蒸发相关联。作为结果，我们的构造也参数化了仿射类型A中的卡齐丹-卢斯滕堡单元。我们推测我们构造的生长图在仿射旗的相对位置上有自然的几何实现，类似于施坦伯格和范·莱文在经典情况下的解释。

Kazhdan-Lusztig left cell preorder and dominance order

Let "$\leq_L$" be the Kazhdan-Lusztig left cell preorder on the symmetric group $S_n$. Let $w\mapsto (P(w),Q(w))$ be the Robinson-Schensted-Knuth correspondence between $S_n$ and the set of standard tableaux with the same shapes. We prove that for any $x,y\in S_n$, $x\leq_L y$ only if $Q(y)\unrhd Q(x)$, where "$\unrhd$" is the dominance (partial) order between standard tableaux. As a byproduct, we generalize an earlier result of Geck by showing that each Kazhdan-Lusztig basis element $C'_w$ can be expressed as a linear combination of some $m_{uv}$ which satisfies that $u\unrhd P(w)^*$, $v\unrh…