Noetherianity up to Symmetry

Draisma, Jan

doi:10.1007/978-3-319-04870-3_2

Cited by 25 publications

(14 citation statements)

References 31 publications

(32 reference statements)

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“…This order is called the Higman order on S * . It is a wellpartial-order by Higman's Lemma (see [12] or, for example, [9]).…”

Section: Inc(n) I -Equivariant Gröbner Basesmentioning

confidence: 99%

“…Fix i ≥ 0 and consider the following partial order on the set of monomials in K[X ] defined by u| Inc(N) i v if there is some π ∈ Inc(N) i such that π(u) divides v. According to [13,Theorem 2.12] (see also [9,Proposition 2.2]) it suffices to show that | Inc(N) i is a well-partial-order.…”

Section: Inc(n) I -Equivariant Gröbner Basesmentioning

confidence: 99%

“…Recent results in algebraic statistics such as the Independent Set Theorem [13,Theorem 4.7] as well as challenging problems on families of varieties, tensors, or representations in spaces of increasing dimension motivate the study of ideals in a polynomial ring K[X ] in a countable set of infinite variables over a field K (see, e.g., [7,8,9,13,16,17]). Often these ideals are invariant under an action of a symmetric group or submonoids of the monoid of strictly increasing functions on the set N of positive integers Inc(N) = {π : N → N | π(i) < π(i + 1) for all i ≥ 1} (see, e.g., [1,13]).…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Equivariant Hilbert series in non-noetherian polynomial rings

Nagel

Römer

2017

Journal of Algebra

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ABSTRACT. We introduce and study equivariant Hilbert series of ideals in polynomial rings in countably many variables that are invariant under a suitable action of a symmetric group or the monoid Inc(N) of strictly increasing functions. Our first main result states that these series are rational functions in two variables. A key is to introduce also suitable submonoids of Inc(N) and to compare invariant filtrations induced by their actions. Extending a result by Hillar and Sullivant, we show that any ideal that is invariant under these submonoids admits a Gröbner basis consisting of finitely many orbits. As our second main result we prove that the Krull dimension and multiplicity of ideals in an invariant filtration grow eventually linearly and exponentially, respectively, and we determine the terms that dominate this growth.

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“…This order is called the Higman order on S * . It is a wellpartial-order by Higman's Lemma (see [12] or, for example, [9]).…”

Section: Inc(n) I -Equivariant Gröbner Basesmentioning

confidence: 99%

Section: Inc(n) I -Equivariant Gröbner Basesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Equivariant Hilbert series in non-noetherian polynomial rings

Nagel

Römer

2017

Journal of Algebra

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“…In [LO11] one can find a set of equations, so-called skew-flattenings, that define the first and second secant variety on an open subset. In [DK11,Dra13] one can find very nice results concerning bounds for the degree of equations that cut out the secant variety set-theoretically. The study of relations among minors is also an interesting, difficult topic [BCV13].…”

Section: Introductionmentioning

confidence: 99%

Secants of minuscule and cominuscule minimal orbits

Manivel

Michałek²

2015

Linear Algebra and its Applications

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Abstract. We study the geometry of the secant and tangential variety of a cominuscule and minuscule variety, e.g. a Grassmannian or a spinor variety. Using methods inspired by statistics we provide an explicit local isomorphism with a product of an affine space with a variety which is the Zariski closure of the image of a map defined by generalized determinants. In particular, equations of the secant or tangential variety correspond to relations among generalized determinants. We also provide a representation theoretic decomposition of cubics in the ideal of the secant variety of any Grassmannian.

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“…1. For other recent results about Sym(N)-Noetherian polynomial algebras see, for instance, articles [2,8,10,11,12] and a survey [7]. 2.…”

Section: Introductionmentioning

confidence: 99%

Symmetric Polynomials and Nonfinitely Generated Sym(ℕ)-Invariant Ideals

Costa¹,

Krasilnikov²

2015

J Math Sci

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Abstract. Let K be a field and let N = {1, 2, . . . }. Let Rn = K[xij | 1 ≤ i ≤ n, j ∈ N] be the ring of polynomials in xij (1 ≤ i ≤ n, j ∈ N) over K. Let Sn = Sym({1, 2, . . . , n}) and Sym(N) be the groups of the permutations of the sets {1, 2, . . . , n} and N, respectively. Then Sn and Sym(N) act on Rn in a natural way: τ (xij) = x τ (i)j and σ(xij) = x iσ(j) for all τ ∈ Sn and σ ∈ Sym(N). Let Rn be the subalgebra of the symmetric polynomials in Rn,In 1992 the second author proved that if char(K) = 0 or char(K) = p > n then every Sym(N)-invariant ideal in Rn is finitely generated (as such). In this note we prove that this is not the case if char(K) = p ≤ n.We also survey some results about Sym(N)-invariant ideals in polynomial algebras and some related results.

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Noetherianity up to Symmetry

Cited by 25 publications

References 31 publications

Equivariant Hilbert series in non-noetherian polynomial rings

Equivariant Hilbert series in non-noetherian polynomial rings

Secants of minuscule and cominuscule minimal orbits

Symmetric Polynomials and Nonfinitely Generated Sym(ℕ)-Invariant Ideals

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