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Topological and Shifting Theoretic Methods in Combinatorics and Algebra
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.).
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: urn:nbn:se:kth:diva-186136ISBN: 978-91-7595-899-6 (print)OAI: oai:DiVA.org:kth-186136DiVA: diva2:925608
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
List of papers
1. Cellular structure for the Herzog–Takayama resolution
Open this publication in new window or tab >>Cellular structure for the Herzog–Takayama resolution
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.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-145027 (URN)10.1007/s10801-014-0524-7 (DOI)000349247100002 ()2-s2.0-84901735112 (Scopus ID)
Note

QC 20140514

Available from: 2014-05-06 Created: 2014-05-06 Last updated: 2017-12-05Bibliographically approved
2. Convex hull of face vectors of colored complexes
Open this publication in new window or tab >>Convex hull of face vectors of colored complexes
2014 (English)In: European journal of combinatorics (Print), ISSN 0195-6698, E-ISSN 1095-9971, Vol. 36, 247-250 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we verify a conjecture by Kozlov [D.N. Kozlov, Convex Hulls of f- and beta-vectors, Discrete Comput. Geom. 18 (1997) 421-431], which describes the convex hull of the set of face vectors of r-colorable complexes on n vertices. As part of the proof we derive a generalization of Turn's graph theorem.

National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-145026 (URN)10.1016/j.ejc.2013.07.004 (DOI)000328869800022 ()2-s2.0-84882950186 (Scopus ID)
Note

QC 20140514

Available from: 2014-05-06 Created: 2014-05-06 Last updated: 2017-12-05Bibliographically approved
3. Clique vectors of k-connected chordal graphs
Open this publication in new window or tab >>Clique vectors of k-connected chordal graphs
2015 (English)In: Journal of combinatorial theory. Series A (Print), ISSN 0097-3165, E-ISSN 1096-0899, Vol. 132, 188-193 p.Article in journal (Refereed) Published
Abstract [en]

The clique vector c(G) of a graph G is the sequence (c(1), c(2), ..., c(d)) in N-d, where c(i) is the number of cliques in G with i vertices and d is the largest cardinality of a clique in G. In this note, we use tools from commutative algebra to characterize all possible clique vectors of k-connected chordal graphs.

Keyword
Clique vectors, Chordal graphs, Graph connectivity
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-145028 (URN)10.1016/j.jcta.2015.01.001 (DOI)000350528900009 ()2-s2.0-84921502318 (Scopus ID)
Note

QC 20150408. Updated from submitted to published.

Available from: 2014-05-06 Created: 2014-05-06 Last updated: 2017-12-05Bibliographically approved
4. Dimension filtration, sequential Cohen-Macaulayness and a new polynomial invariant of graded algebras
Open this publication in new window or tab >>Dimension filtration, sequential Cohen-Macaulayness and a new polynomial invariant of graded algebras
2016 (English)In: Journal of Algebra, ISSN 0021-8693, E-ISSN 1090-266X, Vol. 456, 250-265 p.Article in journal (Refereed) Published
Abstract [en]

Let k be a field and let A be a standard N-graded k-algebra. Using numerical information of some invariants in the primary decomposition of 0 in A, namely the so-called dimension filtration, we associate a bivariate polynomial BW(A;t,w), that we call the Björner-Wachs polynomial, to A.It is shown that the Björner-Wachs polynomial is an algebraic counterpart to the combinatorially defined h-triangle of finite simplicial complexes introduced by Björner & Wachs. We provide a characterisation of sequentially Cohen-Macaulay algebras in terms of the effect of the reverse lexicographic generic initial ideal on the Björner-Wachs polynomial. More precisely, 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 reverse lexicographic generic initial ideal. We conclude by discussing some connections with the Hilbert series of local cohomology modules, extremal Betti numbers and combinatorial Alexander duality.

Place, publisher, year, edition, pages
Academic Press, 2016
Keyword
Sequential Cohen-Macaulayness, Hilbert series, Initial ideal, Extremal Betti numbers
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-186125 (URN)10.1016/j.jalgebra.2016.01.045 (DOI)000375237300011 ()2-s2.0-84960890577 (Scopus ID)
Note

QC 20160509

Available from: 2016-05-02 Created: 2016-05-02 Last updated: 2017-11-30Bibliographically approved
5. Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals
Open this publication in new window or tab >>Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals
2017 (English)In: Journal of the European Mathematical Society (Print), ISSN 1435-9855, E-ISSN 1435-9863, Vol. 19, no 12, 3851-3865 p.Article in journal (Refereed) Published
Abstract [en]

A numerical characterization is given of the h-triangles of sequentially Cohen-Macaulay simplicial complexes. This result determines the number of faces of various dimensions and codimensions that are possible in such a complex, generalizing the classical Macaulay-Stanley theorem to the nonpure case. Moreover, we characterize the possible Betti tables of componentwise linear ideals. A key tool in our investigation is a bijection between shifted multicomplexes of degree <= d and shifted pure. (d - 1)-dimensional simplicial complexes.

Place, publisher, year, edition, pages
European Mathematical Society Publishing House, 2017
Keyword
Simplicial complex, face numbers, Stanley-Reisner rings, sequential Cohen-Macaulayness, componentwise linear ideals
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-186130 (URN)10.4171/JEMS/755 (DOI)000415853100009 ()2-s2.0-85035041682 (Scopus ID)
Funder
Swedish Research Council, 2011-11677-88409-18
Note

QC 20171207

Available from: 2016-05-02 Created: 2016-05-02 Last updated: 2017-12-07Bibliographically approved
6. Connectivity of pseudomanifold graphs from an algebraic point of view
Open this publication in new window or tab >>Connectivity of pseudomanifold graphs from an algebraic point of view
2015 (English)In: Comptes Rendus Mathematiques de l'Academie des Sciences = Mathematical reports of the academy of science, ISSN 0706-1994, Vol. 353, no 12, 1061-1065 p.Article in journal (Refereed) Published
Abstract [en]

The connectivity of graphs of simplicial and polytopal complexes is a classical subject going back at least to Steinitz, and the topic has since been studied by many authors, including Balinski, Barnette, Athanasiadis, and Bjorner. In this note, we provide a unifying approach that allows us to obtain more general results. Moreover, we provide a relation to commutative algebra by relating connectivity problems to graded Betti numbers of the associated Stanley-Reisner rings.

Place, publisher, year, edition, pages
Elsevier, 2015
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-186129 (URN)10.1016/j.crma.2015.09.018 (DOI)000366617600001 ()2-s2.0-84951310212 (Scopus ID)
Note

QC 20160511

Available from: 2016-05-02 Created: 2016-05-02 Last updated: 2017-11-30Bibliographically approved

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