On algorithmically checking whether a Hilbert series comes from a complete intersection

Bøgvad, Rikard; Meyer, Thomas

doi:10.1016/j.jsc.2004.06.001

Cited by 2 publications

(6 citation statements)

References 14 publications

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“…(a) The permutation groupoid on {1, 2, 3} generated by the rank 1 local bijection 1 → 2 is the restriction of the permutation group on {1, 2, 3, 4} generated by the permutation (1, 2) (3,4).…”

Section: 31mentioning

confidence: 99%

“…Let R := (E, ρ), where E := N × {0, 1}, ρ := (N × {0}) 3 ∪ (N × {1}) 3 . Then R has two monomorphic parts, namely N × {0} and N × {1}.…”

Section: Hopf Algebra Structurementioning

confidence: 99%

“…Problem 1.7. Devise some sensible alternative graded algebra structure on K.A(R) which is Cohen-Macaulay whenever the Hilbert series has the appropriate form (by Proposition 4 of [3] such an algebra always exists).…”

Section: Introductionmentioning

confidence: 99%

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Some relational structures with polynomial growth and their associated algebras II. Finite generation.

Pouzet,

Thiéry

2022

CDM

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The profile of a relational structure $R$ is the function $\varphi_R$ which counts for every nonnegative integer $n$ the number, possibly infinite, $\varphi_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $\varphi_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $\mathbb{K}.\mathcal A$ introduced by P. J. Cameron. In a previous paper, we studied the relationship between the properties of a relational structure $R$ and those of its age algebra, particularly when $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups or the rings of quasisymmetric polynomials. The main theorem of this paper characterizes combinatorially when the age algebra is finitely generated in this setting. For tournaments, this boils down to the profile being bounded. We further investigate how far the well known algebraic properties of invariant rings and quasisymmetric polynomials extend to age algebras; notably, we explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra. For a homogeneous structure with a profile bounded by a polynomial, Cameron conjectured in the early eighties that the profile is asymptotically polynomial; Macpherson further conjectured that the age algebra is finitely generated. This was proven recently by Falque and the second author. The combined results support the conjecture that---assuming finite kernel---profiles bounded by a polynomial are asymptotically polynomial, and give hope for a complete characterization of when the age algebra is finitely generated.

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Section: 31mentioning

confidence: 99%

“…Let R := (E, ρ), where E := N × {0, 1}, ρ := (N × {0}) 3 ∪ (N × {1}) 3 . Then R has two monomorphic parts, namely N × {0} and N × {1}.…”

Section: Hopf Algebra Structurementioning

confidence: 99%

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Some relational structures with polynomial growth and their associated algebras II. Finite generation.

Pouzet,

Thiéry

2022

CDM

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“…We do not know for which relational structures having a finite monomorphic decomposition the numerator P can be choosen with non-negative coefficients. A possible approach is to look for some sensible Cohen-Macaulay graded algebra whose Hilbert series is H ϕR (by proposition 4 of [BM04] such a Cohen-Macaulay algebra always exists as soon as P has non-negative coefficients). This is one of our motivations for the upcoming study of the age algebras.…”

Section: 2mentioning

confidence: 99%

“…Problem 0.6. Devise some sensible alternative graded algebra structure on K.A(R) which is Cohen-Macaulay whenever the Hilbert series has the appropriate form (by Proposition 4 of [BM04] such an algebra always exists).…”

Section: Introductionmentioning

confidence: 99%

Some Relational Structures with Polynomial Growth and their Associated Algebras I: Quasi-Polynomiality of the Profile

Pouzet

Thiéry

2013

Electron. J. Combin.

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The profile of a relational structure R is the function ϕ R which counts for every integer n the number ϕ R (n), possibly infinite, of substructures of R induced on the n-element subsets, isomorphic substructures being identified. If ϕ R takes only finite values, this is the Hilbert function of a graded algebra associated with R, the age algebra KA(R), introduced by P. J. Cameron.In this paper we give a closer look at this association, particularly when the relational structure R admits a finite monomorphic decomposition. This setting still encompass well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. We prove that ϕ R is eventually a quasi-polynomial, this supporting the conjecture that, under mild assumptions on R, ϕ R is eventually a quasi-polynomial when it is bounded by some polynomial.

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On algorithmically checking whether a Hilbert series comes from a complete intersection

Cited by 2 publications

References 14 publications

Some relational structures with polynomial growth and their associated algebras II. Finite generation.

Some relational structures with polynomial growth and their associated algebras II. Finite generation.

Some Relational Structures with Polynomial Growth and their Associated Algebras I: Quasi-Polynomiality of the Profile

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