The Partition Complex: an invitation to combinatorial commutative algebra

Adiprasito, Karim; Yashfe, Geva

doi:10.48550/arxiv.2008.01044

Cited by 4 publications

(5 citation statements)

References 16 publications

(25 reference statements)

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“…3. Moreover, by Equation(11) and our previous remarks,h F (∆, x) = n k=0 (h 0 (Γ) + h 1 (Γ) + • • • + h k (Γ)) p F ,n,k (x) h F (Γ, x) = n+1 k=0 h k (Γ) p F ,n+1,k (x) .…”

mentioning

confidence: 72%

“…Thus, one expects that (2) has even better behavior when ∂∆ is a vertex-induced subcomplex of ∆. Indeed, under this assumption, the unimodality of (2) follows for a large family of triangulations ∆ of the ball from [3,Theorem 46] and one can speculate that (2) is also γ-positive when ∆ is flag. In fact, the latter statement can be shown to be equivalent to the equator conjecture, already posed in [21].…”

Section: Introductionmentioning

confidence: 99%

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Symmetric decompositions, triangulations and real-rootedness

Athanasiadis¹,

Tzanaki²

2021

Preprint

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Polynomials which afford nonnegative, real-rooted symmetric decompositions have been investigated recently in algebraic, enumerative and geometric combinatorics. Brändén and Solus have given sufficient conditions under which the image of a polynomial under a certain operator associated to barycentric subdivision has such a decomposition. This paper gives a new proof of their result which generalizes to subdivision operators in the setting of uniform triangulations of simplicial complexes, introduced by the first named author. Sufficient conditions under which these decompositions are also interlacing are described. Applications yield new classes of polynomials in geometric combinatorics which afford nonnegative, real-rooted symmetric decompositions. Some interesting questions in f -vector theory arise from this work.

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“…3. Moreover, by Equation(11) and our previous remarks,h F (∆, x) = n k=0 (h 0 (Γ) + h 1 (Γ) + • • • + h k (Γ)) p F ,n,k (x) h F (Γ, x) = n+1 k=0 h k (Γ) p F ,n+1,k (x) .…”

mentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Symmetric decompositions, triangulations and real-rootedness

Athanasiadis¹,

Tzanaki²

2021

Preprint

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“…We note here that a similar in spirit sequence dealing with the Stanley-Reisner rings of simplicial complexes and their quotients by (generic) linear forms play a central role in the recent works [Adi18,AY20] on partition complexes, and have found several interesting applications.…”

Section: Tropical Deligne Resolution and Cohomology Of Wonderful Comp...mentioning

confidence: 90%

Homology of tropical fans

Amini¹,

Piquerez²

2021

Preprint

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The aim of this paper is to study homological properties of tropical fans and to propose a notion of smoothness in tropical geometry, which goes beyond matroids and their Bergman fans and which leads to an enrichment of the category of smooth tropical varieties.Among the resulting applications, we prove the Hodge isomorphism theorem which asserts that the Chow rings of smooth unimodular tropical fans are isomorphic to the tropical cohomology rings of their corresponding canonical compactifications, and prove a slightly weaker statement for any unimodular fan. We furthermore introduce a notion of shellability for tropical fans and show that shellable tropical fans are smooth and thus enjoy all the nice homological properties of smooth tropical fans. Several other interesting properties for tropical fans are shown to be shellable. Finally, we obtain a generalization, both in the tropical and in the classical setting, of the pioneering work of Feichtner-Yuzvinsky and De Concini-Procesi on the cohomology ring of wonderful compactifications of complements of hyperplane arrangements.The results in this paper form the basis for our subsequent works on Hodge theory for tropical and non-Archimedean varieties.1.1. Tropical smoothness. The definition of tropical smoothness that we propose in this work is homological. In this regard, it is in the spirit of the notion of homology manifold [Čec33,

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“…Face rings are the main object of the paper. Our treatment is standard except for the relative case, in which we follow [AY20]. Fix a field .…”

Section: Basic Notionsmentioning

confidence: 99%

Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles

Adiprasito¹,

Papadakis²,

Petrotou³

2021

Preprint

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We prove the hard Lefschetz property for pseudomanifolds and cycles in any characteristic with respect to an appropriate Artinian reduction. The proof is a combination of Adiprasito's biased pairing theory and a generalization of a formula of Papadakis-Petrotou to arbitary characteristic. In particular, we prove the Lefschetz theorem for doubly Cohen Macaulay complexes, solving a generalization of the g-conjecture due to Stanley. We also provide a simplified presentation of the characteristic 2 case, and generalize it to pseudomanifolds and cycles.Recently, Papadakis and Petrotou [PP20] have built on the same two ideas, and studied anisotropy of Stanley-Reisner rings, a stronger (in the sense of more restrictive) notion

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The Partition Complex: an invitation to combinatorial commutative algebra

Cited by 4 publications

References 16 publications

Symmetric decompositions, triangulations and real-rootedness

Symmetric decompositions, triangulations and real-rootedness

Homology of tropical fans

Anisotropy, biased pairings, and the Lefschetz property for pseudomanifolds and cycles

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