Electric network and Hirota type $4$-simplex maps

Abstract

Bazhanov--Stroganov (4-simplex) maps are set-theoretical solutions to the 4-simplex equation, namely the fourth member of the family of $n$-simplex equations, which are fundamental equations of mathematical physics. In this paper, we develop a method for constructing Bazhanov--Stroganov maps as extensions of tetrahedron maps which are set-theoretical solutions to the Zamolodchikov tetrahedron (3-simplex) equation. We employ this method to construct birarional Bazhanov--Stroganov maps which boil down to the famous electric network and Hirota tetrahedron maps at a certain limit.