Minimax equivalence method: initial ideas, first applications and new concepts

Abstract

In 2005, the author introduced for posets the notion of (min, max)-equivalence (later called minimax equivalence). This equivalence preserves \(\mathbb{Z}\)-equivalence of the corresponding Tits quadratic forms which play an important role in modern representation theory. The minimax equivalence method has been used to solving many classification problems. This method was first applied in the same year by the author together with his PhD student M. V. Styopochkina for classifying all posets with positive Tits quadratic form and all minimal posets with nonpositive Tits form. These results were often cited, but the corresponding publication is virtually inaccessible. The paper provides them (translated into English) and also the author's new ideas about the minimax equivalence method.