  The Chern-Schwartz-MacPherson class (CSM) and the Segre-Schwartz-MacPherson
class (SSM) are deformations of the fundamental class of an algebraic variety.
They encode finer enumerative invariants of the variety than its fundamental
class. In this paper we offer three contributions to the theory of equivariant
CSM/SSM classes. First, we prove an interpolation characterization for CSM
classes of certain representations. This method---inspired by recent works of
Maulik-Okounkov and Gorbounov-Rimanyi-Tarasov-Varchenko---does not require a
resolution of singularities and often produces explicit (not sieve) formulas
for CSM classes. Second, using the interpolation characterization we prove
explicit formulas---including residue generating sequences---for the CSM and
SSM classes of matrix Schubert varieties. Third, we suggest that a stable
version of the SSM class of matrix Schubert varieties will serve as the
building block of equivariant SSM theory, similarly to how the Schur functions
are the building blocks of fundamental class theory. We illustrate these
phenomena, and related stability and (2-step) positivity properties for some
relevant representations.
