Action of W-type operators on Schur functions and Schur Q-functions

In this paper, we investigate a series of W-type differential operators, which appear naturally in the symmetry algebras of KP and BKP hierarchies. In particular, they include all operators in the W-constraints for tau-functions of higher KdV hierarchies that satisfy the string equation. We will give simple uniform formulas for actions of these operators on all ordinary Schur functions and Schur Q-functions. As applications of such formulas, we will give new simple proofs for Alexandrov's conjecture and Mironov-Morozov's formula, which express the Br & eacute;zin-Gross-Witten and Kontsevich-Witten tau-functions as linear combinations of Q-functions with simple coefficients, respectively.