The Dimension of Skew Shifted Young Diagrams, and Projective Characters of the Infinite Symmetric Group

Abstract

Classical Schur P-functions are the particular case of Hall-Littlewood polynomials when the parameter is equal to -1. We introduce factorial (interpolation) analogues of Schur P-functions. A dimension of a skew shifted Young diagram is the number of standard tableaux of the given shape. Also these numbers are equal up to simple factors to the decomposition coefficients of the restriction of an irreducible representation of a spin-symmetric group to a smaller spin-symmetric group. In terms of the factorial Schur P-functions we obtain an explicit formula for the dimension of a skew shifted Young diagram. The main application of this formula is the new derivation of Nazarov's classification of undecomposable projective characters of the infinite symmetric group.