56,422 research outputs found

    Symbolic powers of monomial ideals and Cohen-Macaulay vertex-weighted digraphs

    Full text link
    In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen--Macaulay vertex-weighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen--Macaulay and satisfy Alexander dualityComment: Special volume dedicated to Professor Antonio Campillo, Springer, to appea

    The bases of weighted graphs

    Get PDF
    There are many different mathematical objects (transitive reductions, minimal equivalent digraphs, minimal connected graphs, Hasse diagrams and so on) that are defined on graphs. Although they have different names they correspond to the same object if a weighted graph is defined more generally. The study of such generally defined graphs allows to consider some common properties of the objects, which seem different at the first glance. This article presents a new kind of weighted graphs when the weights of the edges belong to a partially ordered set L. The case, when L is a positive linearly ordered monoid, is considered in more detail. For such L, the weight of a path is equal to the product of the weights of its edges. The exact lower bound of the weights of all paths between two vertices is the distance between these vertices. Graphs with the same set of vertices and equal distance for every pair of vertices form an equivalence class. One can define an order on the set of graphs in the natural way. It is shown that any equivalence class has a smallest element and a non-empty set of maximal elements, the bases. An algorithm is given to find a basis. When an equivalence class has only one basis is also shown

    RMT 555 - PERSEKITARAN PERUNDANGAN APRIL-MAY 06.

    Get PDF
    Abstract. We present a linear time algorithm for computing an implicit linear space representation of a minimum cycle basis (MCB) in weighted partial 2-trees, i.e., graphs of treewidth two. The implicit representation can be made explicit in a running time that is proportional to the size of the MCB. For planar graphs, Borradaile, Sankowski, and Wulff-Nilsen [Min st-cut Oracle for Planar Graphs with Near-Linear Preprocessing Time, FOCS 2010] showed how to compute an implicit O(n log n) space representation of an MCB in O(n log 5 n) time. For the special case of partial 2-trees, our algorithm improves this result to linear time and space. Such an improvement was achieved previously only for outerplanar graphs [Liu and Lu: Minimum Cycle Bases of Weighted Outerplanar Graphs, IPL 110:970–974, 2010]

    Cellularity of KLR and weighted KLRW algebras via crystals

    Full text link
    We prove that the weighted KLRW algebras of finite type, and their cyclotomic quotients, are cellular algebras. The cellular bases are explicitly described using crystal graphs. As a special case, this proves that the KLR algebras of finite type are cellular. As one application, we compute the graded decomposition numbers of the cyclotomic algebras.Comment: 48 pages, many figures, comments welcom
    corecore