Schur--Plucker Geometry of the MDS Locus for Principal-Ideal Codes

Let $g$ be a monic polynomial of degree $r<n$, and let $C_g(n)$ be the coefficient-vector code formed by multiples $ug$ with $\deg(ug)<n$. We study the coefficient-space MDS locus $M_{n,r}$. The companion construction identifies coefficient space with the moduli of cyclic matrix-vector pairs, and the remainder-orbit map embeds it as a smooth complete intersection in the standard big cell of $\operatorname{Gr}(r,n)$. We prove that every normalized maximal Plucker coordinate pulls back, up to sign, to a power of the constant coefficient $A_0$ times a Schur polynomial $S_\kappa(g)=s_\kappa(\Lambda_g)$, where $\kappa\subseteq (n-r)^{r-1}$. Hence the universal MDS polynomial is $$D_{n,r}=A_0\prod_{\kappa\subseteq (n-r)^{r-1}}S_\kappa.$$ This description yields a flat non-MDS boundary over $\mathbb{Z}$, explicit degree and finite-field estimates, and a length filtration governed by sparse multiples. It also gives bad-characteristic criteria on root-multiplicity strata and density-one results on the irreducible stratum. Finally, for $r\ge 3$ and $N\ge r+3$, every nonempty first-failure layer over an algebraically closed field has a dense open non-GRS locus.