On Measures on Partitions Arising in Harmonic Analysis for Linear and Projective Characters of the Infinite Symmetric Group

TL;DR

This paper explores the relationship between z-measures on partitions from harmonic analysis of the infinite symmetric group and their projective counterparts, revealing combinatorial connections via shifted Young diagrams.

Contribution

It introduces combinatorial relations between two families of measures on partitions using the doubling of shifted Young diagrams.

Findings

Identifies combinatorial relations between z-measures and projective measures

Uses doubling of shifted Young diagrams to connect the two measure families

Provides insights into harmonic analysis of the infinite symmetric group

Abstract

The z-measures on partitions originated from the problem of harmonic analysis of linear representations of the infinite symmetric group in the works of Kerov, Olshanski and Vershik (1993, 2004). A similar family corresponding to projective representations was introduced by Borodin (1997). The latter measures live on strict partitions (i.e., partitions with distinct parts), and the z-measures are supported by all partitions. In this note we describe some combinatorial relations between these two families of measures using the well-known doubling of shifted Young diagrams.