Internally Perfect Matroids: In 1977 Stanley conjectured that the h-vector of any matroid is a pure O-sequence, that is, is the Hilbert function of an Artinian monomial algebra whose socle is concentrated in a single degree. Though this conjecture has motivated a great deal of research in combinatorial commutative algebra in the mean time, the general case is still wide open. In the article Internally Perfect Matroids (recently published here) we study a new class of matroids having highly structured internal orders. Indeed, we show that these are precisely the matroids whose internal order is isomorphic to the divisor poset on the standard monomials of an Artinian ideal over a polynomial ring, or equivalently, whose internal orders are certificates for the validity of Stanley’s Conjecture. We show that internally perfect matroids are not contained in any of the classes of matroids for which Stanley’s Conjecture was previously known to hold and prove, up to mild condition, that they form a minor closed class. The many examples in the paper can be verified with the MatroidActivities package and the computations in Macaulay2 that went in to working out the example in the last section of the paper are also worked out here.
PackageCitations: Since my work on the MatroidActivities package for Macaulay2, I have become more involved with more general coding projects for that platform. In particular, I wrote a new package, PackageCitations, to facilitate the citation of Macaulay2 packages in research articles typeset in
LaTeX. The source code and subsequent
bibtex entries are available here. Update: PackageCitations will be included in the next distribution of
Macaulay2 and is already part of the source repository.
Matroid Activities: I wrote a Macaulay2 package for working with external and internal activities in (ordered) matroids. The first version allowed the user to compute the active orders and to check if a given matroid is internally perfect.
In the new version, 0.2, there are many more features. One can create a matroid from either raw combinatorial data, or from a central hyperplane arrangement, a matrix, a graph, a monomial ideal, or a simplicial complex. In addition to being able to compute numerous numerical invariants of a matroid, the user may also compute a number of algebraic structures defined from a matroid including the Stanley-Reisner ring of the independence complex, the Orlik-Solomon algebra, and Chow ring.
In July, CAPTAINS is playing the Festival Internacional Benicàssim (FIB) with the likes The Weekend, Jesus and Mary Chain, and a grip of other big summer rock festival acts. I’m looking forward to looking forward to that show more soon, but this next week I’m setting out on the road to Germany with Ceschi Ramos and Squalloscope. These two are both wonderful performers so if you’re in Frankfurt, Bremen, Hamburg or Lüneburg you’d do well to come out.
The CAPTAINS video for “Touch Me, I’m Driving” was recently released on the ROCKDELUX website. We shot it over two weekends in Gijón with an ace crew lead by the immensely talented David Ferrando Giraut. Have a listen.
Here’s the decidely NSFW video for the Captains’ song Heavy Metal Works. Enjoy.
throttled belttorht: A short piece on swimming, pollen, and the frantic manic panic that takes over every time I’m back to my hometown.
Lawrence oriented matroids: The h-vector of any triangulation of the Lawrence oriented matroid of a representable matroid M is the h-vector of the underlying (unoriented) matroid. We conjecture that the same is true for any oriented matroid and are studying partitions and shellings of such triangulations in order to prove this conjecture. This is ongoing work with Katharina Jochemko.
Your one note: Another repulsion rock number. This one for the moments when you just have to slaughter your white elephants.