Hilbert series of algebras associated to directed graphs and order homology

Retakh, Vladimir; Serconek, Shirlei; Wilson, Robert Lee

doi:10.1016/j.jpaa.2011.10.023

Cited by 4 publications

(19 citation statements)

References 26 publications

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“…This section is a brief digression in order make a connection between our methods and a very good result: Proposition 3.2.1 of [8]. The basic idea of this section is to see just how far one can push the techniques of the previous section without the Koszul hypothesis.…”

Section: Connection To a Theorem Of Retakh Serconek And Wilsonmentioning

confidence: 97%

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Splitting algebras: Koszul, Cohen–Macaulay and numerically Koszul

Kloefkorn

Shelton

2015

Journal of Algebra

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Abstract. We study a finite dimensional quadratic graded algebra RΓ defined from a finite ranked poset Γ. This algebra has been central to the study of the splitting algebras AΓ introduced by Gelfand, Retakh, Serconek and Wilson, [4]. Those algebras are known to be quadratic when Γ satisfies a combinatorial condition known as uniform. A central question in this theory has been: when are the algebras Koszul? We prove that RΓ is Koszul and Γ is uniform if and only if the poset Γ is Cohen-Macaulay. We also show that the cohomology of the order complex of Γ can be identified with certain cohomology groups defined internally to the ring RΓ, HR Γ (n, 0) (introduced in [2]) whenever Γ is Cohen-Macualay. Finally, we settle in the negative the long-standing question: Does numerically Koszul imply Koszul for algebras of the form RΓ.

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Section: Connection To a Theorem Of Retakh Serconek And Wilsonmentioning

confidence: 97%

“…Let Γ be a finite ranked poset with unique minimal element * and strict order <. In the literature this algebra was denoted B(Γ) in [10] and [8] and R(Γ) in [2].…”

Section: Introductionmentioning

confidence: 99%

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Splitting algebras: Koszul, Cohen–Macaulay and numerically Koszul

Kloefkorn

Shelton

2015

Journal of Algebra

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“…The algebras B(Γ) are the dual algebras of the associated graded algebras of A(Γ) under a certain filtration. They have the same Hilbert series as the actual dual of A(Γ) and when it exists, B(Γ) is Koszul if and only if A(Γ) is [7].…”

Section: Theorymentioning

confidence: 99%

“…Proof. We use the numerical Koszulity test from theorem 4.2.1 in [7]. Considering the i = 4 case we know…”

Section: A Layered Graph γ On Nine Vertices With Non-koszul A(γ)mentioning

confidence: 99%

The minimal non-Koszul A(Γ)

Nacin

2017

Communications in Algebra

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The algebras A(Γ), where Γ is a directed layered graph, were first constructed by I. Gelfand, S. Serconek, V. Retakh and R. Wilson. These algebras are generalizations of the algebras Qn, which are related to factorizations of non-commutative polynomials. It was conjectured that these algebras were Koszul. In 2008, T.Cassidy and B.Shelton found a counterexample to this claim, a non-Koszul A(Γ) corresponding to a graph Γ with 18 edges and 11 vertices.We produce an example of a directed layered graph Γ with 13 edges and 9 vertices which produces a non-Koszul A(Γ). We also show this is the minimal example with this property.

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Weakly Cohen–Macaulay posets and a class of finite-dimensional graded quadratic algebras

Kloefkorn

2017

Journal of Algebra

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Abstract. To a finite ranked poset Γ we associate a finite-dimensional graded quadratic algebra RΓ. Assuming Γ satisfies a combinatorial condition known as uniform, RΓ is related to a well-known algebra, the splitting algebra AΓ. First introduced by Gelfand, Retakh, Serconek, and Wilson, splitting algebras originated from the problem of factoring non-commuting polynomials. Given a finite ranked poset Γ, we ask: Is RΓ Koszul? The Koszulity of RΓ is related to a combinatorial topology property of Γ called Cohen-Macaulay. Kloefkorn and Shelton proved that if Γ is a finite ranked cyclic poset, then Γ is Cohen-Macaulay if and only if Γ is uniform and RΓ is Koszul. We define a new generalization of Cohen-Macaulay, weakly Cohen-Macaulay, and we note that this new class includes posets with disconnected open subintervals. We prove: if Γ is a finite ranked cyclic poset, then Γ is weakly Cohen-Macaulay if and only if RΓ is Koszul.

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Hilbert series of algebras associated to directed graphs and order homology

Cited by 4 publications

References 26 publications

Splitting algebras: Koszul, Cohen–Macaulay and numerically Koszul

Splitting algebras: Koszul, Cohen–Macaulay and numerically Koszul

The minimal non-Koszul A(Γ)

Weakly Cohen–Macaulay posets and a class of finite-dimensional graded quadratic algebras

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