Standard monomial bases and geometric consequences for certain rings of invariants

Consider the diagonal action of $S{L}_n(K)$ on the affine space $X=V^{\oplus m}\oplus (V^*)^{\oplus q}$where$V=K^n, K$ an algebraically closed field of arbitrary characteristic and $m,q>n$. We construct a “standard monomial” basis for the ring of invariants $K[X{]}^{S{L}_n(K)}$. As a consequence, we deduce that $K[X{]}^{S{L}_n(K)}$ is Cohen-Macaulay. We also present the first and second fundamental theorems for $S{L}_n(K)$-actions.