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Cellular structure for the Herzog–Takayama resolution
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2014 (English)In: Journal of Algebraic Combinatorics, ISSN 0925-9899, E-ISSN 1572-9192Article in journal (Refereed) Published
Abstract [en]

Herzog and Takayama constructed an explicit resolution for the ideals with a regular linear quotient. These ideals include all matroidal and stable ideals. The resolutions of matroidal and stable ideals are known to be cellular. In this note, we show that the Herzog–Takayama resolution is also cellular.

Place, publisher, year, edition, pages
2014.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-145027DOI: 10.1007/s10801-014-0524-7ISI: 000349247100002Scopus ID: 2-s2.0-84901735112OAI: oai:DiVA.org:kth-145027DiVA: diva2:715774
Note

QC 20140514

Available from: 2014-05-06 Created: 2014-05-06 Last updated: 2017-12-05Bibliographically approved
In thesis
1. On Face Vectors and Resolutions
Open this publication in new window or tab >>On Face Vectors and Resolutions
2014 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consist of the following three papers.

  • Convex hull of face vectors of colored complexes. In this paper we verify a conjecture by Kozlov (Discrete ComputGeom18(1997) 421–431), which describes the convex hull of theset of face vectors ofr-colorable complexes onnvertices. As partof the proof we derive a generalization of Turán’s graph theorem.
  • Cellular structure for the Herzog–Takayama Resolution. Herzog and Takayama constructed explicit resolution for the ide-als in the class of so called ideals with a regular linear quotient.This class contains all matroidal and stable ideals. The resolu-tions of matroidal and stable ideals are known to be cellular. Inthis note we show that the Herzog–Takayama resolution is alsocellular.
  • Clique Vectors ofk-Connected Chordal Graphs. The clique vectorc(G)of a graphGis the sequence(c1,c2,...,cd)inNd, whereciis the number of cliques inGwithivertices anddis the largest cardinality of a clique inG. In this note, we usetools from commutative algebra to characterize all possible cliquevectors ofk-connected chordal graphs.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. ix, 21 p.
Series
TRITA-MAT. MA, ISSN 1401-2278 ; 2014:07
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-145029 (URN)978-91-7595-153-9 (ISBN)
Presentation
2014-05-30, Rum 3721, Matematik, Lindstedtsvägen 25, plan 7, KTH, Stockholm, 13:15 (English)
Opponent
Supervisors
Note

QC 20140513

Available from: 2014-05-13 Created: 2014-05-06 Last updated: 2014-05-14Bibliographically approved
2. Topological and Shifting Theoretic Methods in Combinatorics and Algebra
Open this publication in new window or tab >>Topological and Shifting Theoretic Methods in Combinatorics and Algebra
2016 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of six papers related to combinatorics and commutative algebra.

In Paper A, we use tools from topological combinatorics to describe the minimal free resolution of ideals with a so called regular linear quotient. Our result generalises the pervious results by Mermin and by Novik, Postnikov & Sturmfels.

In Paper B, we describe the convex hull of the set of face vectors of coloured simplicial complexes. This generalises the Turan Graph Theorem and verifies a conjecture by Kozlov from 1997.

In Paper C, we use algebraic shifting methods to characterise all possible clique vectors of k-connected chordal graphs.

In Paper D, to every standard graded algebra we associate a bivariate polynomial that we call the Björner-Wachs polynomial. We show that this invariant provides an algebraic counterpart to the combinatorially defined h-triangle of simplicial complexes. Furthermore, we show that a graded algebra is sequentially Cohen-Macaulay if and only if it has a stable Björner-Wachs polynomial under passing to the generic initial ideal.

In Paper E, we give a numerical characterisation of the h-triangle of sequentially Cohen-Macaulay simplicial complexes; answering an open problem raised by Björner & Wachs in 1996. This generalise the Macaulay-Stanley Theorem. Moreover, we characterise the possible Betti diagrams of componentwise linear ideals.

In Paper F, we use algebraic and topological tools to provide a unifying approach to study the connectivity of manifold graphs. This enables us to obtain more general results.

Place, publisher, year, edition, pages
KTH Royal Institute of Technology, 2016. 152 p.
Series
TRITA-MAT-A, 2016:02
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-186136 (URN)978-91-7595-899-6 (ISBN)
Public defence
2016-06-07, F3, Lindstedtsvägen 26, Stockholm, 12:30 (English)
Opponent
Supervisors
Note

QC 20160516

Available from: 2016-05-16 Created: 2016-05-02 Last updated: 2016-05-16Bibliographically approved

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