  We introduce certain torus-equivariant classes on permutohedral varieties
which we call "tautological classes of matroids" as a new geometric framework
for studying matroids. Using this framework, we unify and extend many recent
developments in matroid theory arising from its interaction with algebraic
geometry. We achieve this by establishing a Chow-theoretic description and a
log-concavity property for a 4-variable transformation of the Tutte polynomial,
and by establishing an exceptional Hirzebruch-Riemann-Roch-type formula for
permutohedral varieties that translates between K-theory and Chow theory.
