.

Verify the identity for Slater determinants,

$$\langle \xi_1,...,\xi_N \vert\nu_1,...,\nu_N\rangle_A \equiv \fra... ...t{N!}}\sum_p {\rm sign}(p) \psi_{\nu_1}( \xi_{p(1)})...\psi_{\nu_N}(\xi_{p(N)})$$

$$= \frac{1}{\sqrt{N!}}\sum_p {\rm sign}(p) \psi_{\nu_{p(1)}}( \xi_1)...\psi_{\nu_{p(N)}}(\xi_N).$$ (3.1)

by explicit calculation for $N=3$ Fermions.