KdV and mKdV Hierarchies and Schur Q-functions

We prove two conjectures on the Korteweg-de Vries (KdV) and modified KdV (mKdV) hierarchies and Schur Q-functions presented by Yamada. The first one is that the functions defined by Sato and Mori in 1980 coincide with Schur Q-functions indexed by even or odd strict partitions. Mizukawa, Nakajima, and Yamada gave an expression for this function using symmetric functions and Littlewood-Richardson coefficients. We prove that this expression coincides with the Schur Q-function by using the formula of Lascoux, Leclerc, and Thibon. The second one is that Schur Q-functions indexed by strict partitions which have odd parts form a basis for the space of Hirota polynomials of the KdV hierarchy, and that Schur Q-functions indexed by strict partitions which have even parts form a basis for the space of Hirota polynomials of the mKdV hierarchy. This conjecture is verified by rewriting the generating series of the KdV and mKdV hierarchies using the techniques of symmetric functions.