标题： 单峰集上保序函数和逆序函数的独子表示理论 显示英文标题

标题： Representation theory of monoids consisting of order-preserving functions and order-reversing functions on an n-set

摘要： 设$\operatorname{OD}_{n}$为集合$\{1,\ldots,n\}$上所有保序函数和逆序函数的独异点。 我们描述了单子代数$\Bbbk\operatorname{OD}_{n}$的一个箭图表示，其中$\Bbbk$是一个特征不是 2 的域。 我们证明该箭图包含两条直线路径，一条有$n-1$个顶点，另一条有$n$个顶点，并且所有连续箭头的复合都等于$0$。 作为证明的一部分，我们得到了所有由诱导左Schützenberger模之间的同态的完整描述，这些模属于$\Bbbk\operatorname{OD}_{n}$。 我们还定义$\operatorname{COD}_{n}$为$\operatorname{OD}_{n}$的覆盖，并在保序和反序常函数之间引入了人为的区别。 我们证明了$\operatorname{COD}_{n}\simeq\operatorname{O}_{n}\rtimes\mathbb{Z}_{2}$，其中$\operatorname{O}_{n}$是集合$\{1,\ldots,n\}$上所有保序函数的独异点。 此外，如果 $\Bbbk$是一个特征不是 $2$的域，我们证明了 $\Bbbk\operatorname{COD}_{n}\simeq\Bbbk\operatorname{O}_{n}\times\Bbbk\operatorname{O}_{n}$。 作为推论，我们得出 $\Bbbk\operatorname{COD}_{n}$的箭图由两条具有 n 个顶点的直线路径组成，并且所有连续箭头的复合都等于 $0$。 显示英文摘要

摘要： Let $\operatorname{OD}_{n}$ be the monoid of all order-preserving functions and order-reversing functions on the set $\{1,\ldots,n\}$. We describe a quiver presentation for the monoid algebra $\Bbbk\operatorname{OD}_{n}$ where $\Bbbk$ is a field whose characteristic is not 2. We show that the quiver consists of two straightline paths, one with $n-1$ vertices and one with $n$ vertices, and that all compositions of consecutive arrows are equal to $0$. As part of the proof we obtain a complete description of all homomorphisms between induced left Sch\"utzenberger modules of $\Bbbk\operatorname{OD}_{n}$. We also define $\operatorname{COD}_{n}$ to be a covering of $\operatorname{OD}_{n}$ with an artificial distinction between order-preserving and order-reversing constant functions. We show that $\operatorname{COD}_{n}\simeq\operatorname{O}_{n}\rtimes\mathbb{Z}_{2}$ where $\operatorname{O}_{n}$ is the monoid of all order-preserving functions on the set $\{1,\ldots,n\}$. Moreover, if $\Bbbk$ is a field whose characteristic is not $2$ we prove that $\Bbbk\operatorname{COD}_{n}\simeq\Bbbk\operatorname{O}_{n}\times\Bbbk\operatorname{O}_{n}$. As a corollary, we deduce that the quiver of $\Bbbk\operatorname{COD}_{n}$ consists of two straightline paths with n vertices, and that all compositions of consecutive arrows are equal to $0$.