Singular Loci of Ladder Determinantal Varieties and Schubert Varieties

We relate certain ladder determinantal varieties (associated to one-sided ladders) to certain Schubert varieties in SL(n)/Q, for a suitable n and a suitable parabolic subgroup Q, and we determine the singular loci of these varieties. We state a conjecture on the irreducible components of the singular locus of a Schubert variety in the flag variety, which is a refinement of the conjecture of Lakshmibai and Sandhya (Proc. Indian Acad. Sci. Math. Sci.100 (1990), 45–52). We prove the conjecture for a certain class of Schubert varieties.