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

2025/07/20 by Itamar Stein, Stein, Itamar
Computer Science · Mathematics · #20M20 #20M25 #20M30 #Advanced Algebra and Logic #Algebraic structures and combinatorial models #FOS: Mathematics #Group Theory (math.GR) #Representation Theory (math.RT) #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.2507.14873

openalex publication_date 2025/07/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let OD_n be the monoid of all order-preserving functions and order-reversing functions on the set \1,…,n\. We describe a quiver presentation for the monoid algebra \BbbkOD_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 \BbbkOD_n. We also define COD_n to be a covering of OD_n with an artificial distinction between order-preserving and order-reversing constant functions. We show that COD_n\simeqO_n\rtimesℤ₂ where O_n is the monoid of all order-preserving functions on the set \1,…,n\. Moreover, if \Bbbk is a field whose characteristic is not 2 we prove that \BbbkCOD_n≃\BbbkO_n×\BbbkO_n. As a corollary, we deduce that the quiver of \BbbkCOD_n consists of two straightline paths with n vertices, and that all compositions of consecutive arrows are equal to 0.

Citations

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Related

• The representation theory of the monoid of all partial functions on a set and related monoids as EI-category algebras
• Representation theory of order-related monoids of partial functions as locally trivial category algebras
• The monoid of families of quiver representations
• Topology and monoid representations II: left regular bands of groups and Hsiao's monoid of ordered G-partitions
• The Hall algebra and the composition monoid