92 research outputs found
A matroid-friendly basis for the quasisymmetric functions
A new Z-basis for the space of quasisymmetric functions (QSym, for short) is
presented. It is shown to have nonnegative structure constants, and several
interesting properties relative to the space of quasisymmetric functions
associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, and
Reiner. In particular, for loopless matroids, this basis reflects the grading
by matroid rank, as well as by the size of the ground set. It is shown that the
morphism F is injective on the set of rank two matroids, and that
decomposability of the quasisymmetric function of a rank two matroid mirrors
the decomposability of its base polytope. An affirmative answer is given to the
Hilbert basis question raised by Billera, Jia, and Reiner.Comment: 25 pages; exposition tightened, typos corrected; to appear in the
Journal of Combinatorial Theory, Series
Coloring Complexes and Combinatorial Hopf Monoids
We generalize the notion of coloring complex of a graph to linearized
combinatorial Hopf monoids. These are a generalization of the notion of
coloring complex of a graph. We determine when a combinatorial Hopf monoid has
such a construction, and discover some inequalities that are satisfied by the
quasisymmetric function invariants associated to the combinatorial Hopf monoid.
We show that the collection of all such coloring complexes forms a
combinatorial Hopf monoid, which is the terminal object in the category of
combinatorial Hopf monoids with convex characters. We also study several
examples of combinatorial Hopf monoids.Comment: 37 pages, 5 figure
A quasisymmetric function for matroids
AbstractA new isomorphism invariant of matroids is introduced, in the form of a quasisymmetric function. This invariant: •defines a Hopf morphism from the Hopf algebra of matroids to the quasisymmetric functions, which is surjective if one uses rational coefficients;•is a multivariate generating function for integer weight vectors that give minimum total weight to a unique base of the matroid;•is equivalent, via the Hopf antipode, to a generating function for integer weight vectors which keeps track of how many bases minimize the total weight;•behaves simply under matroid duality;•has a simple expansion in terms of P-partition enumerators;•is a valuation on decompositions of matroid base polytopes. This last property leads to an interesting application: it can sometimes be used to prove that a matroid base polytope has no decompositions into smaller matroid base polytopes. Existence of such decompositions is a subtle issue arising from the work of Lafforgue, where lack of such a decomposition implies that the matroid has only a finite number of realizations up to scalings of vectors and overall change-of-basis
The Tchebyshev transforms of the first and second kind
We give an in-depth study of the Tchebyshev transforms of the first and
second kind of a poset, recently discovered by Hetyei. The Tchebyshev transform
(of the first kind) preserves desirable combinatorial properties, including
Eulerianess (due to Hetyei) and EL-shellability. It is also a linear
transformation on flag vectors. When restricted to Eulerian posets, it
corresponds to the Billera, Ehrenborg and Readdy omega map of oriented
matroids. One consequence is that nonnegativity of the cd-index is maintained.
The Tchebyshev transform of the second kind is a Hopf algebra endomorphism on
the space of quasisymmetric functions QSym. It coincides with Stembridge's peak
enumerator for Eulerian posets, but differs for general posets. The complete
spectrum is determined, generalizing work of Billera, Hsiao and van
Willigenburg.
The type B quasisymmetric function of a poset is introduced. Like Ehrenborg's
classical quasisymmetric function of a poset, this map is a comodule morphism
with respect to the quasisymmetric functions QSym.
Similarities among the omega map, Ehrenborg's r-signed Birkhoff transform,
and the Tchebyshev transforms motivate a general study of chain maps. One such
occurrence, the chain map of the second kind, is a Hopf algebra endomorphism on
the quasisymmetric functions QSym and is an instance of Aguiar, Bergeron and
Sottile's result on the terminal object in the category of combinatorial Hopf
algebras. In contrast, the chain map of the first kind is both an algebra map
and a comodule endomorphism on the type B quasisymmetric functions BQSym.Comment: 33 page
Valuations for matroid polytope subdivisions
We prove that the ranks of the subsets and the activities of the bases of a
matroid define valuations for the subdivisions of a matroid polytope into
smaller matroid polytopes.Comment: 19 pages. 2 figures; added section 6 + other correction
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