Stable states and representations of the infnite symmetric group

TL;DR

This paper introduces stable representations for the infinite symmetric group, providing a complete classification and linking them to admissible representations of the double-group, resolving longstanding questions about their type.

Contribution

It defines the new class of stable representations and characterizes them completely for the infinite symmetric group, connecting them to admissible double-group representations.

Findings

Stable representations are fully classified for ${rak S}_{N}$.

All stable representations are of type ${ m II}$.

The family of stable representations coincides with components of admissible double-group representations.

Abstract

We introduce the notion of stable representations, -- it is a new class of the representations of a certain class of groups which defined with positive definite functions which generalize the classical notion of the characters (or trace). We give the complete description of this class for infinite symmetric group ${\frak S}_{\Bbb N}$. It happened the family of the stable representations of ${\frak S}_{\Bbb N}$ coincides with the set of representations of the components (left or right) of the admissible representations of the double-group in the sense of Ol'shanski. In particular all these are representation of type ${\rm II}$ which was open question for many years.