Representation theory of the infinite symmetric group and Pfaffian point processes

TL;DR

This paper develops Pfaffian point processes linked to the harmonic analysis of the infinite symmetric group, providing explicit formulas for their kernels using Whittaker functions, with parallels to symplectic ensembles in Random Matrix Theory.

Contribution

It introduces a new family of Pfaffian point processes for the infinite symmetric group with explicit kernel formulas involving classical special functions.

Findings

Correlation functions are Pfaffians with matrix kernels.

Explicit kernel formulas are derived using Whittaker functions.

The structure parallels symplectic ensembles in Random Matrix Theory.

Abstract

We construct a family of Pfaffian point processes relevant for the harmonic analysis on the infinite symmetric group. The correlation functions of these processes are representable as Pfaffians with matrix valued kernels. We give explicit formulae for the matrix valued kernels in terms of the classical Whittaker functions. The obtained formulae have the same structure as that arising in the study of symplectic ensembles of Random Matrix Theory. The paper is an extended version of the author's talk at Fall 2010 MSRI Random Matrix Theory program.