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This paper uses dualities between facet ideal theory and Stanley-Reisner theory to show that the facet ideal of a simplicial tree is sequentially Cohen-Macaulay. The proof involves showing that the Alexander dual (or the cover dual, as we call it here) of a simplicial tree is a componentwise linear ideal. We conclude with additional combinatorial properties… (More)

- Sara Faridi
- J. Comb. Theory, Ser. A
- 2005

In this paper we study simplicial complexes as higher dimensional graphs in order to produce algebraic statements about their facet ideals. We introduce a large class of square-free monomial ideals with Cohen-Macaulay quotients, and a criterion for the Cohen-Macaulayness of facet ideals of simplicial trees. Along the way, we generalize several concepts from… (More)

Tanisaki introduced generating sets for the defining ideals of the schematic intersections of the closure of conjugacy classes of nilpotent matrices with the set of diagonal matrices. These ideals are naturally labeled by integer partitions. Given such a partition λ, we define several methods to produce a reduced generating set for the associated ideal Iλ.… (More)

viii

- Massimo Caboara, Sara Faridi, Peter Selinger
- J. Symb. Comput.
- 2007

We generalize the concept of a cycle from graphs to simplicial complexes. We show that a simplicial cycle is either a sequence of facets connected in the shape of a circle, or is a cone over such a structure. We show that a simplicial tree is a connected cycle-free simplicial complex, and use this characterization to produce an algorithm that checks in… (More)

We find a minimal generating set for the defining ideal of the schematic intersection of the set of diagonal matrices with the closure of the conjugacy class of a nilpotent matrix indexed by a hook partition. The structure of this ideal allows us to compute its minimal free resolution and give an explicit description of the graded Betti numbers, and study… (More)

- Diane Maclagan, Rekha R. Thomas, +4 authors Tony Puthenpurakal

Index 130 Bibliography 133 Introduction This book is based on six lectures and tutorials that were prepared for a workshop in computational commutative algebra at the Harish Chandra Research Institute (HRI) at Allahabad, India in December 2003. The workshop was aimed at graduate students and was conducted as part of the conference on Commutative Algebra and… (More)

- Kia Dalili, Sara Faridi, Will Traves
- Discrete Mathematics
- 2008

Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebraic properties of the edge ideal from the deck of G, that is, from the collection of subgraphs obtained by removing a vertex from G. These properties include the Krull dimension, the Hilbert function, and all the graded Betti numbers i,j where j <n. We also state… (More)

- Emma L. Connon, Sara Faridi
- J. Comb. Theory, Ser. A
- 2013

We present an algorithm that checks in polynomial time whether a simplicial complex is a tree. We also present an efficient algorithm for checking whether a simplicial complex is grafted. These properties have strong algebraic implications for their corresponding facet ideals.