4,447 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
A free subalgebra of the algebra of matroids
This paper is an initial inquiry into the structure of the Hopf algebra of
matroids with restriction-contraction coproduct. Using a family of matroids
introduced by Crapo in 1965, we show that the subalgebra generated by a single
point and a single loop in the dual of this Hopf algebra is free.Comment: 19 pages, 3 figures. Accepted for publication in the European Journal
of Combinatorics. This version incorporates a few minor corrections suggested
by the publisher
The Free product of Matroids
We introduce a noncommutative binary operation on matroids, called free
product. We show that this operation respects matroid duality, and has the
property that, given only the cardinalities, an ordered pair of matroids may be
recovered, up to isomorphism, from its free product. We use these results to
give a short proof of Welsh's 1969 conjecture, which provides a progressive
lower bound for the number of isomorphism classes of matroids on an n-element
set.Comment: 5 pages, 1 figure. Accepted for publication in the European Journal
of Combinatorics. See also arXiv:math.CO/040902
A unique factorization theorem for matroids
We study the combinatorial, algebraic and geometric properties of the free
product operation on matroids. After giving cryptomorphic definitions of free
product in terms of independent sets, bases, circuits, closure, flats and rank
function, we show that free product, which is a noncommutative operation, is
associative and respects matroid duality. The free product of matroids and
is maximal with respect to the weak order among matroids having as a
submatroid, with complementary contraction equal to . Any minor of the free
product of and is a free product of a repeated truncation of the
corresponding minor of with a repeated Higgs lift of the corresponding
minor of . We characterize, in terms of their cyclic flats, matroids that
are irreducible with respect to free product, and prove that the factorization
of a matroid into a free product of irreducibles is unique up to isomorphism.
We use these results to determine, for K a field of characteristic zero, the
structure of the minor coalgebra of a family of matroids that
is closed under formation of minors and free products: namely, is
cofree, cogenerated by the set of irreducible matroids belonging to .Comment: Dedicated to Denis Higgs. 25 pages, 3 figures. Submitted for
publication in the Journal of Combinatorial Theory (A). See
arXiv:math.CO/0409028 arXiv:math.CO/0409080 for preparatory work on this
subjec
Products of Linear Forms and Tutte Polynomials
Let \Delta be a finite sequence of n vectors from a vector space over any
field. We consider the subspace of \operatorname{Sym}(V) spanned by \prod_{v
\in S} v, where S is a subsequence of \Delta. A result of Orlik and Terao
provides a doubly indexed direct sum of this space. The main theorem is that
the resulting Hilbert series is the Tutte polynomial evaluation
T(\Delta;1+x,y). Results of Ardila and Postnikov, Orlik and Terao, Terao, and
Wagner are obtained as corollaries.Comment: Minor changes. Accepted for publication in European Journal of
Combinatoric
The Way of Love: Practicing an Irigarayan Ethic
Merwe, W.L. [Promotor]van der Olthuis, J.H. [Promotor]Halsema, J.M. [Copromotor
Modular elimination in matroids and oriented matroids
We introduce a new axiomatization of matroid theory that requires the
elimination property only among modular pairs of circuits, and we present a
cryptomorphic phrasing thereof in terms of Crapo's axioms for flats. This new
point of view leads to a corresponding strengthening of the circuit axioms for
oriented matroids.Comment: 6 pages; v2: text modified in order to better reflect the published
version, references update
An Algebra of Pieces of Space -- Hermann Grassmann to Gian Carlo Rota
We sketch the outlines of Gian Carlo Rota's interaction with the ideas that
Hermann Grassmann developed in his Ausdehnungslehre of 1844 and 1862, as
adapted and explained by Giuseppe Peano in 1888. This leads us past what Rota
variously called 'Grassmann-Cayley algebra', or 'Peano spaces', to the Whitney
algebra of a matroid, and finally to a resolution of the question "What,
really, was Grassmann's regressive product?". This final question is the
subject of ongoing joint work with Andrea Brini, Francesco Regonati, and
William Schmitt.
The present paper was presented at the conference "The Digital Footprint of
Gian-Carlo Rota: Marbles, Boxes and Philosophy" in Milano on 17 Feb 2009. It
will appear in proceedings of that conference, to be published by Springer
Verlag.Comment: 28 page
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