Groups generated by involutions, numberings of posets, and central measures

TL;DR

This paper introduces a new class of countable groups generated by involutions acting on monotonic numberings of posets, exploring their properties, invariant measures, and connections to generalized infinite symmetric groups.

Contribution

It defines a novel class of groups based on poset numberings, extending Coxeter groups, and investigates their invariant measures and representation theory.

Findings

Defined new groups generated by involutions on poset numberings

Connected invariant measures to generalized infinite symmetric group representations

Discussed properties and problems related to infinite groups of this type

Abstract

We define a new class of countable groups, which are defined by its action on the set of monotonic numberings (diagrams) of an arbitrary finite or countable partial ordered set (poset). These groups are generated by the set of involutions? and in the case of finite posets could be considered as generalization of Coxeter's symmetric groups. We discuss the problems concerned to infinite groups jf this type, in particular the problem of the descripton of invariant measures on the space of numberings (central measures)with respect to actions of those groups. The probelms are tightly connected with the new theory of representations of the generalizations of infinite symmetric group.