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Decompositions of polytopes and valuations are objects of
fundamental
geometric interest. Decompositions of matroid polytopes in
particular
have made several prominent recent appearances, in tropical
geometry
and in constructions of good compactifications of certain
varieties;
and several important matroid and polymatroid invariants are
valuations.
We construct explicit bases for the modules of matroids and
polymatroids on a fixed ground set modulo relations given by
decompositions, and their duals, the modules of valuative
functions.
The basis elements for matroid polytopes are essentially the
Schubert
matroids. These results carry over to the unlabelled setting,
dual to
valuative matroid invariants, and to the setting of the
associated
graded modules in the grading by dimension, dual to valuations
that
are zero on non-full-dimensional polytopes. This work is joint with Harm Derksen. |