  We consider the Schur-positivity of monomial immanants of Jacobi-Trudi
matrices, in particular whether a non-negative coefficient of the trivial Schur
function implies non-negative coefficients for other Schur functions in said
immanants. We prove that this true for hook-shape Schur functions using
combinatorial methods in a representation theory setting. Our main theorem
proves that hook-shape immanant characters can be written as finite
non-negative integer sums of Stanley-Stembridge characters, and provides an
explicit combinatorial formula for these sums. This resolves a special case of
a longstanding conjecture of Stanley and Stembridge that posits such a sum
exists for all immanant characters. We also provide several simplifications for
computing immanant characters, and several corollaries applying the main result
to cases where the coefficient of the trivial Schur function in monomial
immanants of Jacobi-Trudi matrices is known to be non-negative.
