On the Structure of Set-Theoretic Polygon Equations

• Veröffentlicht in: Symmetry, Integrability and Geometry: Methods and Applications
• Datum: 2024
• ISSN: 1815-0659
• 1. Verfasser: Müller-Hoissen, Folkert
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут математики НАН України 2024
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/212256
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: On the Structure of Set-Theoretic Polygon Equations. Folkert Müller-Hoissen. SIGMA 20 (2024), 051, 30 pages

Polygon equations generalize the prominent pentagon equation in very much the same way as simplex equations generalize the famous Yang-Baxter equation. In particular, they appeared as ''cocycle equations'' in Street's category theory associated with oriented simplices. Whereas the ( − 1)-simplex equation can be regarded as a realization of the higher Bruhat order (, − 2), the -gon equation is a realization of the higher Tamari order (, − 2). The latter and its dual ~(, − 2), associated with which is the dual -gon equation, have been shown to arise as suborders of (N, N−2) via a ''three-color decomposition''. There are two different reductions of (, − 2) and ~(, − 2), to ( − 1, − 3), respectively ~( − 1, − 3). In this work, we explore the reductions of (dual) polygon equations, which yield relations between solutions of neighboring (dual) polygon equations. We also elaborate (dual) polygon equations in this respect explicitly up to the octagon equation.