Representation theory of monoids consisting of order-preserving functions and order-reversing functions on an n-set
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912687176286208 |
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| author | Stein, Itamar |
| author_facet | Stein, Itamar |
| contents | 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 $0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_14873 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Representation theory of monoids consisting of order-preserving functions and order-reversing functions on an n-set Stein, Itamar Representation Theory Group Theory 20M20, 20M25, 20M30 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 $0$. |
| title | Representation theory of monoids consisting of order-preserving functions and order-reversing functions on an n-set |
| topic | Representation Theory Group Theory 20M20, 20M25, 20M30 |
| url | https://arxiv.org/abs/2507.14873 |