Polarizations and hook partitions

In this paper, we relate combinatorial conditions for polarizations of powers of the graded maximal ideal with rank conditions on submodules generated by collections of Young tableaux. We apply discrete Morse theory to the hypersimplex resolution introduced by Batzies–Welker to show that the L-complex of Buchsbaum and Eisenbud for powers of the graded maximal ideal is supported on a CW-complex. We then translate the “spanning tree condition” of Almousa–Fløystad–Lohne characterizing polarizations of powers of the graded maximal ideal into a condition about which sets of hook tableaux span a certain Schur module. As an application, we give a complete combinatorial characterization of polarizations of so-called “restricted powers” of the graded maximal ideal.     

Looking out for stable syzygy bundles

We study (slope-)stability properties of syzygy bundles on a projective space PN given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle. Restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve. From this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two-dimensional complete intersection ring.     

Stanley-Reisner ideals whose powers have finite length cohomologies

We introduce a class of Stanley-Reisner ideals called a generalized complete intersection, which is characterized by the property that all the residue class rings of powers of the ideal have FLC. We also give a combinatorial characterization of such ideals.

     

The toric ideal of a graphic matroid is generated by quadrics

Describing minimal generating sets of toric ideals is a well-studied and difficult problem. Neil White conjectured in 1980 that the toric ideal associated to a matroid is generated by quadrics corresponding to single element symmetric exchanges. We give a combinatorial proof of White’s conjecture for graphic matroids.     

Chorded complexes and a necessary condition for a monomial ideal to have a linear resolution

In this paper we extend one direction of Fröberg?s theorem on a combinatorial classification of quadratic monomial ideals with linear resolutions. We do this by generalizing the notion of a chordal graph to higher dimensions with the introduction of d-chorded and orientably-d-cycle-complete simplicial complexes. We show that a certain class of simplicial complexes, the d-dimensional trees, correspond to ideals having linear resolutions over fields of characteristic 2 and we also give a necessary combinatorial condition for a monomial ideal to be componentwise linear over all fields.     

Combinatorial proofs of some determinantal identities

In this paper, we give combinatorial proofs of some determinantal identities. In fact, we give a combinatorial proof of a theorem of R. P. Stanley regarding the enumeration of paths in acyclic digraphs along with some interesting applications. We also give an almost visual proof of a recent result of Oliver Knill, namely ‘The generalized Cauchy–Binet Theorem.’     

Certain variants of multipermutohedron ideals

Multipermutohedron ideals have rich combinatorial properties. An explicit combinatorial formula for the multigraded Betti numbers of a multipermutohedron ideal and their Alexander duals are known. Also, the dimension of the Artinian quotient of an Alexander dual of a multipermutohedron ideal is the number of generalized parking functions. In this paper, monomial ideals which are certain variants of multipermutohedron ideals are studied. Multigraded Betti numbers of these variant monomial ideals and their Alexander duals are obtained. Further, many interesting combinatorial properties of multipermutohedron ideals are extended to these variant monomial ideals.     

Semilattice orders on the homomorphic images of the Rédei semigroup

The combinatorial simple principal ideal semigroups generated by two elements were described by L. Megyesi and G. Pollák. The ‘most general’ among them is called the Rédei semigroup. The ‘most special’ combinatorial simple principal ideal semigroup generated by two elements is the bicyclic semigroup. D. B. McAlister determined the compatible semilattice orders on the bicyclic semigroup. Our aim is to study the compatible semilattice orders on the homomorphic images of the Rédei semigroup. We prove that there are four compatible total orders on these semigroups. We show that on the Rédei semigroup, the total orders are the only compatible semilattice orders. Moreover, on each proper homomorphic image of the Rédei semigroup, we give a compatible semilattice order which is not a total order. Communicated by Mária B. Szendrei     

On the index and the order of quasi-regular implicit systems of differential equations

This paper is mainly devoted to the study of the differentiation index and the order for quasi-regular implicit ordinary differential algebraic equation (DAE) systems. We give an algebraic definition of the differentiation index and prove a Jacobi-type upper bound for the sum of the order and the differentiation index. Our techniques also enable us to obtain an alternative proof of a combinatorial bound proposed by Jacobi for the order.As a consequence of our approach we deduce an upper bound for the Hilbert–Kolchin regularity and an effective ideal membership test for quasi-regular implicit systems. Finally, we prove a theorem of existence and uniqueness of solutions for implicit differential systems.     

Combinatorial 3-Manifolds with Transitive Cyclic Symmetry

In this article we give combinatorial criteria to decide whether a transitive cyclic combinatorial d-manifold can be generalized to an infinite family of such complexes, together with an explicit construction in the case that such a family exists. In addition, we substantially extend the classification of combinatorial 3-manifolds with transitive cyclic symmetry up to 22 vertices. Finally, a combination of these results is used to describe new infinite families of transitive cyclic combinatorial manifolds and in particular a family of neighborly combinatorial lens spaces of infinitely many distinct topological types.     

Embedding graphs in surfaces

The basic topological facts about closed curves in $\text{R}$² and triangulability of surfaces are used to prove the folk theorem that any surface embedding of a graph is combinatorial. A basic technical lemma for this proof (a version of what it means to apply scissors to an embedded graph) is then used to give a rigourous definition of the combinatorial boundary of a face and also to introduce a combinatorial definition of equivalence of embeddings. This latter definition is on the one hand easily seen to correspond correctly to the natural topological notion of equivalence, and on the other hand to give equivalence classes in 1-1 correspondence with the classes coming from combinatorial definitions of earlier authors.     

Linear resolutions of quadratic monomial ideals

We study the minimal free resolution of a quadratic monomial ideal in the case where the resolution is linear. First, we focus on the squarefree case, namely that of an edge ideal. We provide an explicit minimal free resolution under the assumption that the graph associated with the edge ideal satisfies specific combinatorial conditions. In addition, we construct a regular cellular structure on the resolution. Finally, we extend our results to non-squarefree ideals by means of polarization.     

Mathias–Prikry and Laver type forcing; summable ideals,coideals, and +-selective filters

We study the Mathias–Prikry and the Laver type forcings associated with filters and coideals. We isolate a crucial combinatorial property of Mathias reals, and prove that Mathias–Prikry forcings with summable ideals are all mutually bi-embeddable. We show that Mathias forcing associated with the complement of an analytic ideal always adds a dominating real. We also characterize filters for which the associated Mathias–Prikry forcing does not add eventually different reals, and show that they are countably generated provided they are Borel. We give a characterization of \({\omega}\)-hitting and \({\omega}\)-splitting families which retain their property in the extension by a Laver type forcing associated with a coideal.     

Dimension and enumeration of primitive ideals in quantum algebras

In this paper, we study the primitive ideals of quantum algebras supporting a rational torus action. We first prove a quantum analogue of a Theorem of Dixmier; namely, we show that the Gelfand-Kirillov dimension of primitive factors of various quantum algebras is always even. Next we give a combinatorial criterion for a prime ideal that is invariant under the torus action to be primitive. We use this criterion to obtain a formula for the number of primitive ideals in the algebra of 2×n quantum matrices that are invariant under the action of the torus. Roughly speaking, this can be thought of as giving an enumeration of the points that are invariant under the induced action of the torus in the “variety of 2×n quantum matrices”. The first author thanks NSERC for its generous support. This research was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme held at the University of Edinburgh, by a Marie Curie European Reintegration Grant within the 7th European Community Framework Programme and by Leverhulme Research Interchange Grant F/00158/X.