α-Pfaffian, pfaffian point process and shifted Schur measure

For any complex number αand any even-size skew-symmetric matrix B, we define a generalization pfα(B) of the pfaffian pf(B) which we call the α-pfaffian. The α-pfaffian is a pfaffian analogue of the α-determinant studied in [T. Shirai and Y. Takahashi, J. Funct. Anal. 205 (2003) 414-463] and [D. Vere-Jones, Linear Algebra Appl. 111 (1988) 119-124]. 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 obtained in [S. Matsumoto, Correlation functions of the shifted Schur measure, math.CO/0312373] for the correlation function of the shifted Schur measure.