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 $0$.
Categories: math.RT math.GR