The V/L recursion for Macdonald's 7th Variation Schur polynomials

We generalize and prove the recursive relation $$S_{\lambda }(V)=\sum _{L\subseteq V\text{ line}}{S}_{\lambda }(VL)$$ conjectured by I. G. Macdonald for his “7th variation” of the Schur functions. This variation is a family of polynomials over a finite field that mimic the (straight and skew) Schur polynomials using powers of the Frobenius.