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"K-theoretic" analog of Postnikov-Shapiro algebra distinguishes graphs
Stockholm University, Faculty of Science, Department of Mathematics.
Stockholm University, Faculty of Science, Department of Mathematics.
2017 (English)In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 148, p. 316-332Article in journal (Refereed) Published
Abstract [en]

In this paper we study a filtered "K-theoretical" analog of a graded algebra associated to any loopless graph G which was introduced in \cite{PS}. We show that two such filtered algebras are isomorphic if and only if their graphs are isomorphic. We also study a large family of filtered generalizations of the latter graded algebra which includes the above "K-theoretical" analog.

Place, publisher, year, edition, pages
2017. Vol. 148, p. 316-332
Keywords [en]
Spanning forests and trees, Commutative algebras, Filtered algebras
National Category
Discrete Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:su:diva-132988DOI: 10.1016/j.jcta.2017.01.001ISI: 000394723900011OAI: oai:DiVA.org:su-132988DiVA, id: diva2:955998
Available from: 2016-08-28 Created: 2016-08-28 Last updated: 2018-04-07Bibliographically approved
In thesis
1. On a class of commutative algebras associated to graphs
Open this publication in new window or tab >>On a class of commutative algebras associated to graphs
2016 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

In 2004 Alexander Postnikov and Boris Shapiro introduced a class of commutative algebras for non-directed graphs. There are two main types of such algebras, algebras of the first type count spanning trees and algebras  of the second type count spanning forests. These algebras have a number of interesting properties including an explicit formula for their Hilbert series. In this thesis we mainly work with the second type of algebras, we discover more properties of the original algebra and construct a few generalizations. In particular we prove that the algebra counting forests depends only on graphical matroid of the graph and converse. Furthermore, its "K-theoretic" filtration reconstructs the whole graph. We introduse $t$ labelled algebras of a graph, their Hilbert series contains complete information about the Tutte polynomial of the initial graph. Finally we introduce similar algebras for hypergraphs. To do this, we define spanning forests and trees of a hypergraph and the corresponding "hypergraphical" matroid.

Place, publisher, year, edition, pages
Stockholm: Department of Mathematics, Stockholm University, 2016. p. 40
National Category
Discrete Mathematics Algebra and Logic
Identifiers
urn:nbn:se:su:diva-132987 (URN)
Presentation
2016-09-05, Sal 14, hus 5, Kräftriket, Stockholm, 13:00 (English)
Opponent
Supervisors
Available from: 2016-11-03 Created: 2016-08-28 Last updated: 2016-11-03Bibliographically approved
2. 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
Research subject
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)
Opponent
Supervisors
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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