Given a partition $\lambda$, the Schur polynomials can be defined, among many other ways, as $$S_{\lambda}(\xi_1,\dots,\xi_a)=\frac{\det\left(\xi_i^{\lambda_j+a-j}\right)_{i,j=1}^a}{\det\left(\xi_i^{a-j}\right)_{i,j=1}^a}.$$ Let $(n^c)=(n,\dots,n)\vdash cn$ be a short-hand notation. Now, define the polynomials $$\Psi(n^c,a):=s_{(n^c)}(\xi_1,\dots,\xi_a).$$
I would like to ask:
QUESTION. Assume $0<c\leq a$. Is this identity true? \begin{align*} \Psi(n^c,a)\cdot\Psi((n-2)^c,a+1) &=\Psi((n-1)^c,a)\cdot \Psi((n-1)^c,a+1) \\ &-\Psi((n-1)^{c-1},a)\cdot \Psi((n-1)^{c+1},a+1). \end{align*}
