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

Let ${\mathrm{OD}}_n$ be the monoid of all order-preserving functions and order-reversing functions on the set $\{1,\dots ,n\}$. We describe a quiver presentation for the monoid algebra $\mathbb{k}{\mathrm{OD}}_n$ where $\mathbb{k}$ 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 Schützenberger modules of $\mathbb{k}{\mathrm{OD}}_n$. We also define ${\mathrm{COD}}_n$ to be a covering of ${\mathrm{OD}}_n$ with an artificial distinction between order-preserving and order-reversing constant functions. We show that ${\mathrm{COD}}_n\simeq {\mathrm{O}}_n⋊{\mathbb{Z}}_2$ where ${\mathrm{O}}_n$ is the monoid of all order-preserving functions on the set $\{1,\dots ,n\}$. Moreover, if $\mathbb{k}$ is a field whose characteristic is not $2$ we prove that $\mathbb{k}{\mathrm{COD}}_n\simeq \mathbb{k}{\mathrm{O}}_n\times \mathbb{k}{\mathrm{O}}_n$. As a corollary, we deduce that the quiver of $\mathbb{k}{\mathrm{COD}}_n$ consists of two straightline paths with n vertices, and that all compositions of consecutive arrows are equal to $0$.