发表机构
Shamoon College of Engineering（沙蒙工程学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文给出从 N-tableau 直接读出其对应 Catalan 幺半群元素的可视化方法，通过引入定位与满核概念，证明满 N-tableaux 与 Catalan 幺半群一一对应。
AI 中文摘要
由 Abram 和 Reutenauer 引入的 stylic 幺半群 $\mathrm{Styl}_n$ 是 plactic 幺半群关于关系 $x^2=x$ 的商，其元素由 $N$-tableaux 表示。Volkov 证明了 $\{0,1,\ldots,n\}$ 上保序且序递减的自映射构成的 Catalan 幺半群 $\mathrm{Cat}_n$ 是 $\mathrm{Styl}_n$ 的商。然而，商映射定义在生成元上，从 $N$-tableau 中看不出它对应于 $\mathrm{Cat}_n$ 中的哪个映射。本文给出一种简单的可视化方法，直接从 $N$-tableau 读出该映射及其主要性质。新的要点在于我们不坚持将 $N$-tableau 画成经典 Young 表：我们允许每一行的条目相对于下一行发生平移，只要每个条目仍位于一个较小条目之上。我们称此为定位（positioning），并证明从任意定位读出的列词与通常的列词是 plactic 等价的；因此任何定位都可用来计算商映射。我们研究紧致定位（tight positioning），其中每个条目尽可能向右推，并定义 $N$-tableau 的满核（full core）：每一列中从底行开始按连续值上升的部分。若 $N$-tableau 等于其满核，则称其为满的。我们证明取满核不改变在 $\mathrm{Cat}_n$ 中的像，满 $N$-tableaux 与 $\mathrm{Cat}_n$ 一一对应，并展示如何直接从满 $N$-tableau 读出相应映射。
英文摘要
The stylic monoid $\mathrm{Styl}_n$, introduced by Abram and Reutenauer, is the quotient of the plactic monoid by the relations $x^2=x$, and its elements are represented by $N$-tableaux. Volkov showed that the Catalan monoid $\mathrm{Cat}_n$ of order-preserving, order-decreasing self-maps of $\{0,1,\ldots,n\}$ is a quotient of $\mathrm{Styl}_n$. However, the quotient map is defined on generators, and it is not apparent how to see, from an $N$-tableau, the map in $\mathrm{Cat}_n$ it corresponds to. In this paper we give a simple visual way to read off this map, and some of its main properties, from the $N$-tableau. The new ingredient is that we do not insist on drawing an $N$-tableau as a classical Young tableau: we allow the entries of each row to be shifted relative to the row below, as long as each entry stays above a smaller one. We call this a positioning, and prove that the column word read from any positioning is plactically equivalent to the usual column word; so every positioning can be used to compute the quotient map. We work with the tight positioning, in which each entry is pushed as far right as possible, and define the full core of an $N$-tableau: the part of each column that climbs by consecutive values from the bottom row. We call an $N$-tableau full if it equals its full core. We prove that passing to the full core does not change the image in $\mathrm{Cat}_n$, that full $N$-tableaux are in bijection with $\mathrm{Cat}_n$, and we show how to read the corresponding map directly off a full $N$-tableau.
Comments25 pages, 18 figures
