On tetrahedron type equations associated with $B_3,C_3, F_4$ and $H_3$

TL;DR

This paper explores generalizations of the tetrahedron equation, a 3D analogue of the Yang-Baxter equation, focusing on Coxeter groups B_3, C_3, F_4, and H_3, providing remarks and insights on these extensions.

Contribution

It offers miscellaneous remarks on extending the tetrahedron equation to specific Coxeter groups beyond A_3, including non-crystallographic groups, supplementing existing theoretical frameworks.

Findings

Remarks on generalizations along B_3, C_3, F_4, and H_3

Connections to Coxeter groups and their properties

Insights into the structure of tetrahedron equations

Abstract

Tetrahedron equation is a three dimensional analogue of the Yang-Baxter equation. It allows a formulation in terms of the Coxeter group $A_3$. This short note includes miscellaneous remarks on the generalizations along $B_3, C_3, F_4$ and the non-crystallographic Coxeter group $H_3$. It is a supplement to the author's talk in the online workshop, Combinatorial Representation Theory and Connections with Related Fields, at RIMS, Kyoto University in November 2021.