I write for both an element of the Weyl group and the corresponding point in . Let be a Schubert variety. Then is an inclusion of representations. The latter is generated by the -weight space, where is the highest root.
Suppose , where is the reflection corresponding to . Then the connecting and lies in (think of the corresponding SL2), so the -weight space lies in . Since this generates , we have .
Therefore if , then is singular. This includes some of the first examples of singular Schubert varieties, for example the B2 singular Schubert variety and one of the A3 ones.
