Singular loci of cominuscule Schubert varieties

Abstract

Let X=G/P be a cominuscule rational homogeneous variety. Equivalently, X admits the structure of a compact Hermitian symmetric space. I give a uniform description (that is, independent of type) of the irreducible components of the singular locus of a Schubert variety Y⊂X in terms of representation theoretic data. The result is based on a recent characterization of the Schubert varieties using an integer a≥0 and a marked Dynkin diagram. Corollaries include: (1) the variety is smooth if and only if a=0; (2) if G is of type ADE, then the singular locus occurs in codimension at least 3.