Applications of Minor-Summation Formula II. Pfaffians and Schur Polynomials

The purpose of this paper is, to establish, by extensive use of the minor summation formula of pfaffians exploited in (Ishikawa, Okada, and Wakayama, J. Algebra 183, 193-216) certain new generating functions involving Schur polynomials which have a product representation. This generating function gives an extension of the Littlewood formula. During the course of the proof we develop some techniques for computing sub-Pfaffians of a given skew-symmetric matrix. After the proof we present an open problem which generalizes our formula.