Towards the characterization of minimum weight codewords of Schubert codes

A conjectural formula for the minimum weight of Schubert codes was conjectured by Ghorpade in 2000. This was established by Xiang in 2008. In 2018, Ghorpade and Singh provided a new proof of this conjecture. Moreover, they also conjectured that the minimum weight codewords of the Schubert codes $C_{\text{a}}(\mathrm{\ell },m)$ are given by the so-called \emph{Schubert decomposable codewords}. We prove the validity of the conjecture for all Schubert varieties ${\mathrm{\Omega }}_{\text{a}}(\mathrm{\ell },V_m)$ for all but finitely many values of $q$.