Representation theory of monoids consisting of order-preserving functions and order-reversing functions on an n-set
Itamar Stein
TL;DR
This work analyzes the representation theory of the monoids OD_n and COD_n of order-preserving and order-reversing functions on [n]. It develops a quiver presentation for the monoid algebra over a field with characteristic not equal to 2, showing OD_n has a quiver consisting of two straight-line components with n−1 and n vertices and square-zero consecutive arrows; COD_n is shown to be a covering and a semidirect product O_n ⋊ Z_2, with its algebra isomorphic to k O_n × k O_n, yielding a quiver with two n-vertex paths. The approach combines induced Schützenberger modules, skeletal category techniques, and explicit morphism computations to obtain complete Hom-space descriptions and a concrete decomposition into projective modules. The COD_n results deliver a clean parallel to OD_n, including a direct-product algebra decomposition, and together these results give explicit, computable quiver data and Morita-invariant insights for these transformation monoids. Overall, the paper advances a systematic, category-theoretic method for deriving quiver presentations and module decompositions for transformation monoids with rich order-structure.
Abstract
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 Schützenberger 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 $0$.