37 research outputs found
Coxeter submodular functions and deformations of Coxeter permutahedra
We describe the cone of deformations of a Coxeter permutahedron, or
equivalently, the nef cone of the toric variety associated to a Coxeter
complex. This family of polytopes contains polyhedral models for the
Coxeter-theoretic analogs of compositions, graphs, matroids, posets, and
associahedra. Our description extends the known correspondence between
generalized permutahedra, polymatroids, and submodular functions to any finite
reflection group.Comment: Minor edits. To appear in Advances of Mathematic
K-theoretic positivity for matroids
Hilbert polynomials have positivity properties under favorable conditions. We
establish a similar "K-theoretic positivity" for matroids. As an application,
for a multiplicity-free subvariety of a product of projective spaces such that
the projection onto one of the factors has birational image, we show that a
transformation of its K-polynomial is Lorentzian. This partially answers a
conjecture of Castillo, Cid-Ruiz, Mohammadi, and Montano. As another
application, we show that the h*-vector of a simplicially positive divisor on a
matroid is a Macaulay vector, affirmatively answering a question of Speyer for
a new infinite family of matroids
Polyhedral and Tropical Geometry of Flag Positroids
A flag positroid of ranks on is a
flag matroid that can be realized by a real matrix such that
the minors of involving rows are
nonnegative for all . In this paper we explore the polyhedral
and tropical geometry of flag positroids, particularly when
is a sequence of consecutive numbers. In
this case we show that the nonnegative tropical flag variety
TrFl equals the nonnegative flag Dressian
FlDr, and that the points of TrFl
FlDr give rise to coherent subdivisions of the
flag positroid polytope into flag positroid
polytopes. Our results have applications to Bruhat interval polytopes: for
example, we show that a complete flag matroid polytope is a Bruhat interval
polytope if and only if its -dimensional faces are Bruhat interval
polytopes. Our results also have applications to realizability questions. We
define a positively oriented flag matroid to be a sequence of positively
oriented matroids which is also an oriented flag
matroid. We then prove that every positively oriented flag matroid of ranks
is realizable.Comment: 40 page
K-theoretic Tutte polynomials of morphisms of matroids
We generalize the Tutte polynomial of a matroid to a morphism of matroids via
the K-theory of flag varieties. We introduce two different generalizations, and
demonstrate that each has its own merits, where the trade-off is between the
ease of combinatorics and geometry. One generalization recovers the Las Vergnas
Tutte polynomial of a morphism of matroids, which admits a corank-nullity
formula and a deletion-contraction recursion. The other generalization does
not, but better reflects the geometry of flag varieties.Comment: 27 pages; minor revisions. To appear in JCT