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