LCM Lattices and Stanley Depth: A First Computational Approach

Ichim, Bogdan; Katthän, Lukas; Moyano‐Fernández, Julio José

doi:10.1080/10586458.2015.1005257

Cited by 8 publications

(7 citation statements)

References 18 publications

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“…In several computational experiments, we have classified all lcm-lattices of ideals I with up to five generators and found that the projective dimension and the Stanley projective dimension of S/I coincide for these lattices; this is presented in [IKMF16].…”

Section: Introductionmentioning

confidence: 99%

Stanley depth and the lcm-lattice

Ichim

Katthän

Moyano‐Fernández

2017

Journal of Combinatorial Theory, Series A

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ABSTRACT. In this paper we show that the Stanley depth, as well as the usual depth, are essentially determined by the lcm-lattice. More precisely, we show that for quotients I/J of monomial ideals J ⊂ I, both invariants behave monotonic with respect to certain maps defined on their lcm-lattice. This allows simple and uniform proofs of many new and known results on the Stanley depth. In particular, we obtain a generalization of our result on polarization presented in [IKMF15]. We also obtain a useful description of the class of all monomial ideals with a given lcm-lattice, which is independent from our applications to the Stanley depth.

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Section: Introductionmentioning

confidence: 99%

Stanley depth and the lcm-lattice

Ichim

Katthän

Moyano‐Fernández

2017

Journal of Combinatorial Theory, Series A

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“…A result in this direction could, for example, be used to study the following question. It is motivated by the observation in [IKMF14a] that the Stanley projective dimension and the (usual) projective dimension coincide for all ideals with up to five generators. In view of Lemma 6.3 and Theorem 5.6 we offer the following conjecture.…”

Section: Discussion and Open Problemsmentioning

confidence: 99%

“…This complements several similar results [KSF14; HJY08; IKMF14b] and in particular implies the Stanley conjecture for ideals with up to five generators. The latter has also been obtained by different methods in [IKMF14a]. Then we turn to the case of six generators.…”

Section: Introductionmentioning

confidence: 97%

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Stanley depth and simplicial spanning trees

Katthän

2015

J Algebr Comb

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Abstract. We show that for proving the Stanley conjecture, it is sufficient to consider a very special class of monomial ideals. These ideals (or rather their lcm lattices) are in bijection with the simplicial spanning trees of skeletons of a simplex.We apply this result to verify the Stanley conjecture for quotients of monomial ideals with up to six generators. For seven generators we obtain a partial result. IntroductionLet K be a field, S an N n -graded K-algebra and M a finitely generated Z n -graded S-module. The Stanley depth of M, denoted sdepth M, is a combinatorial invariant of M related to a conjecture of Stanley from 1982 [Sta82, Conjecture 5.1] (nowadays called the Stanley conjecture), which states that depth M ≤ sdepth M. We refer the reader to [Pou+09] for an introduction to the subject and to the survey by Herzog [Her13] for a comprehensive account of the known results. Most of the research concentrates on the particular case where S is a polynomial ring and M is either a monomial ideal I ⊂ S or a quotient S/I. In the present paper we will also work in this setting.The main result of the present paper is that for proving the Stanley conjecture for M = S/I or M = I, it is sufficient to consider certain very special ideals. These ideals (more precisely their lcm lattices) are in bijection with certain simplicial complexes which we call stoss complexes (short for Spanning Tree Of a Skeleton of a Simplex ):These complexes can be seen as high dimensional analogues of trees. They have already been studied by Kalai [Kal83] in 1983, and more recently by Duval, Klivans and Martin [DKM09] and others. For k ∈ N we denote by B(k) the boolean algebra on k generators, i.e. the set of all subsets of

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“…Further, both ideals have only five generators, so their Stanley projective dimensions coincide with the respective projective dimensions (Ichim et al 2016). In particular, their Betti posets are nonisomorphic and their Stanley projective dimensions differ.…”

Section: H(s/imentioning

confidence: 99%