  In the literature there are several determinant formulas for Schur functions:
the Jacobi-Trudi formula, the dual Jacobi-Trudi formula, the Giambelli formula,
the Lascoux-Pragacz formula, and the Hamel-Goulden formula, where the
Hamel-Goulden formula implies the others. In this paper we use an identity
proved by Bazin in 1851 to derive determinant identities involving Macdonald's
9th variation of Schur functions. As an application we prove a determinant
identity for factorial Schur functions conjectured by Morales, Pak, and Panova.
We also obtain a generalization of the Hamel-Goulden formula, which contains a
result of Jin, and prove a converse of the Hamel-Goulden theorem and its
generalization.
