Alpha-Pfaffian, pfaffian point process and shifted Schur measure

TL;DR

This paper introduces the alpha-pfaffian, a generalization of the pfaffian linked to the alpha-determinant, explores its properties, and applies it to define point processes and analyze the shifted Schur measure.

Contribution

It defines the alpha-pfaffian, studies its properties, and connects it to point processes and the shifted Schur measure with new explicit pfaffian formulas.

Findings

Defined the alpha-pfaffian as a pfaffian analogue of the alpha-determinant.

Derived formulas and positivity results for alpha-pfaffians.

Provided a linear algebraic proof for the pfaffian expression of the shifted Schur measure's correlation function.

Abstract

For any complex number $\alpha$ and any even-size skew-symmetric matrix $B$, we define a generalization $\pfa{\alpha}(B)$ of the pfaffian $\pf(B)$ which we call the $\alpha$-pfaffian. The $\alpha$-pfaffian is a pfaffian analogue of the $\alpha$-determinant. It gives the pfaffian at $\alpha=-1$. We give some formulas for $\alpha$-pfaffians and study the positivity. Further we define point processes determined by the $\alpha$-pfaffian. Also we provide a linear algebraic proof of the explicit pfaffian expression for the correlation function of the shifted Schur measure.