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Postnikov-Shapiro Algebras, Graphical Matroids and their generalizations
Stockholm University, Faculty of Science, Department of Mathematics.
##### Abstract [en]

In this paper we consider the original and different generalizations of Postnikov-Shapiro algebra which enumerate forests and trees of graphs, see~\cite{PSh}. Our main result is that the algebra counting forests depends only on graphical matroid and converse. Also we generalize algebras for a hypergraph. For this, we define spanning forests and trees of a hypergraph and the corresponding "hypergraphical" matroid. We present 3 different equivalent definitions of spanning forests and trees, which can be read independently from other parts of the paper.

##### Keywords [en]
Tutte polynomial, graphical matroid, spanning forest
##### National Category
Discrete Mathematics
Mathematics
##### Identifiers
OAI: oai:DiVA.org:su-154517DiVA, id: diva2:1194304
Available from: 2018-03-30 Created: 2018-03-30 Last updated: 2018-04-07Bibliographically approved
##### In thesis
1. Around power ideals: From Fröberg's conjecture to zonotopal algebra
Open this publication in new window or tab >>Around power ideals: From Fröberg's conjecture to zonotopal algebra
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

In this thesis we study power algebras, which are quotient of polynomial rings by power ideals. We will study Hilbert series of such ideals and their other properties. We consider two important special cases, namely, zonotopal ideals and generic ideals. Such ideals have a lot combinatorial properties.

In the first chapter we study zonotopal ideals, which were defined and used in several earlier publications. The most important works are by F.Ardila and A.Postnikov and by O.Holtz and A.Ron. These papers originate from different sources, the first source is homology theory, the second one is the theory of box splines. We study quotient algebras by these ideals; these algebras have a nice interpretation for their Hilbert series, as specializations of their Tutte polynomials. There are two important subclasses of these algebras, called unimodular and graphical. The graphical algebras were defined by A.Postnikov and B.Shapiro. In particular, the external algebra of a complete graph is exactly the algebra generated by the Bott-Chern forms of the corresponding complete flag variety. One of the main results of the thesis is a characterization of external algebras. In fact, for the case of graphical and unimodular algebras we prove that external algebras are in one-to-one correspondence with graphical and regular matroids, respectively.

In the second chapter we study Hilbert series of generic ideals. By a generic ideal we mean an ideal generated by forms from some class, whose coefficients belong to a Zariski-open set. There are two main classes to consider: the first class is when we fix the degrees of generators; the famous Fröberg's conjecture gives the expected Hilbert series of such ideals; the second class is when an ideal is generated by powers of generic linear forms. There are a few partial results on Fröberg's conjecture, namely, when the number of variables is at most three. In both classes the Hilbert series is known in the case when the number of generators is at most (n+1). In both cases we construct a lot of examples when the degree of generators are the same and the Hilbert series is the expected one.

##### Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2018. p. 58
##### National Category
Discrete Mathematics Algebra and Logic
Mathematics
##### Identifiers
urn:nbn:se:su:diva-154903 (URN)978-91-7797-244-0 (ISBN)978-91-7797-245-7 (ISBN)
##### Public defence
2018-05-25, sal 14, hus 5, Kräftriket, Roslagsvägen 101, Stockholm, 13:00 (English)
##### Note

At the time of the doctoral defense, the following papers were unpublished and had a status as follows: Paper 1: Manuscript. Paper 4: Manuscript. Paper 5: Manuscript. Paper 7: Manuscript. Paper 8: Manuscript.

Available from: 2018-05-02 Created: 2018-04-07 Last updated: 2018-05-07Bibliographically approved

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##### By organisation
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##### On the subject
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