  We give a direct proof of the equivalence between the Giambelli and Pieri
type formulas for Hall-Littlewood functions using Young's raising operators,
parallel to joint work with Buch and Kresch for the Schubert classes on
isotropic Grassmannians. We prove several closely related mirror identities
enjoyed by the Giambelli polynomials, which lead to new recursions for Schubert
classes. The raising operator approach is applied to obtain tableau formulas
for Hall-Littlewood functions, theta polynomials, and related Stanley symmetric
functions. Finally, we introduce the notion of a skew element w of the
hyperoctahedral group and identify the set of reduced words for w with the set
of standard k-tableaux on a skew Young diagram.
