  In a classical-type flag variety, we consider a Schubert variety associated
to a vexillary (signed) permutation, and establish a combinatorial formula for
the Hilbert-Samuel multiplicity of a point on such a Schubert variety. The
formula is expressed in terms of excited Young diagrams, and extends results
for Grassmannians due to Krattenthaler, Lakshmibai-Raghavan-Sankaran, and for
the maximal isotropic (symplectic and orthogonal) Grassmannians to
Ghorpade-Raghavan, Raghavan-Upadhyay, Kreiman, and Ikeda-Naruse. We also
provide a new proof of a theorem of Li-Yong in the type A vexillary case. The
main ingredient is an isomorphism between certain neighborhoods of fixed
points, known as Kazhdan-Lusztig varieties, which, in turn, relies on a direct
sum embedding previously used by Anderson-Fulton to relate vexillary loci to
Grassmannian loci.
