Finite generation of symmetric ideals

Aschenbrenner, Matthias; Hillar, Christopher J.

doi:10.1090/s0002-9947-07-04116-5

Cited by 79 publications

(137 citation statements)

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“…In proving the Noetherianity of R, it was shown that a symmetric ideal I has a special, finite set of generators called a minimal Gröbner basis. However, the more basic question of whether I is always cyclic (already asked by Josef Schicho [4]) was left unanswered in [1]. Our result addresses a generalization of this important issue.…”

mentioning

confidence: 62%

“…. }, although as remarked in [1], this is not really a restriction. In this case, S X is naturally identified with S ∞ , the permutations of the positive integers, and…”

Section: Theorem 1 For Every Positive Integer N There Are Symmetricmentioning

confidence: 91%

“…Aschenbrenner and Hillar recently proved [1] that all symmetric ideals are finitely generated over R[S X ]. They were motivated by finiteness questions in chemistry [3] and algebraic statistics [2].…”

Section: Christopher J Hillar and Troels Windfeldt (Communicated By mentioning

confidence: 99%

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Minimal generators for symmetric ideals

Hillar

Windfeldt

2008

Proc. Amer. Math. Soc.

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Abstract. Let R = K[X] be the polynomial ring in infinitely many indeterminates X over a field K, and let S X be the symmetric group of X. The group S X acts naturally on R, and this in turn gives R the structure of a module over the group ring R[S X ]. A recent theorem of Aschenbrenner and Hillar states that the module R is Noetherian. We address whether submodules of R can have any number of minimal generators, answering this question positively.Let R = K[X] be the polynomial ring in infinitely many indeterminates X over a field K. Write S X (resp. S N ) for the symmetric group of X (resp. {1, . . . , N}) and R[S X ] for its (left) group ring, which acts naturally on R.Aschenbrenner and Hillar recently proved [1] that all symmetric ideals are finitely generated over R[S X ]. They were motivated by finiteness questions in chemistry [3] and algebraic statistics [2]. In proving the Noetherianity of R, it was shown that a symmetric ideal I has a special, finite set of generators called a minimal Gröbner basis. However, the more basic question of whether I is always cyclic (already asked by Josef Schicho [4]) was left unanswered in [1]. Our result addresses a generalization of this important issue. Theorem 1. For every positive integer n, there are symmetric ideals of R generated by n polynomials which cannot have fewer than n R[S X ]-generators.In what follows, we work with the set X = {x 1 , x 2 , x 3 , . . .}, although as remarked in [1], this is not really a restriction. In this case, S X is naturally identified with S ∞ , the permutations of the positive integers, and σx i = x σi for σ ∈ S ∞ .Let M be a finite multiset of positive integers and let i 1 , . . . , i k be the list of its distinct elements, arranged so thatFor instance, the multiset M = {1, 1, 1, 2, 3, 3} has type λ(M ) = (3, 2, 1). Multisets are in bijection with monomials of R. Given M , we can construct the monomial:Conversely, given a monomial, the associated multiset is the set of indices appearing in it, along with multiplicities. The action of S ∞ on monomials coincides with the

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mentioning

confidence: 62%

“…. }, although as remarked in [1], this is not really a restriction. In this case, S X is naturally identified with S ∞ , the permutations of the positive integers, and…”

Section: Theorem 1 For Every Positive Integer N There Are Symmetricmentioning

confidence: 91%

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Minimal generators for symmetric ideals

Hillar

Windfeldt

2008

Proc. Amer. Math. Soc.

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“…For n = 1 this theorem was proved by Cohen [3] in 1967 and rediscovered independently by Aschenbrenner and Hillar [1] in 2007; for an arbitrary positive n this was proved by Cohen [4] in 1987 and rediscovered independently by Hillar and Sullivant [13] in 2012. Cohen's results were motivated by the finite basis problem for identities of metabelian groups and the results of Aschenbrenner, Hillar and Sullivant by applications to chemistry and algebraic statistics.…”

Section: Introductionmentioning

confidence: 95%

“…The following theorem was proved by Draisma [6] in 2010; it solves a problem arising from applications to algebraic statistics and chemistry posed in [1].…”

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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Codimension and projective dimension up to symmetry

Nagel

Nguyen

et al. 2019

Mathematische Nachrichten

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Symmetric ideals in increasingly larger polynomial rings that form an ascending chain are investigated. We focus on the asymptotic behavior of codimensions and projective dimensions of ideals in such a chain. If the ideals are graded it is known that the codimensions grow eventually linearly. Here this result is extended to chains of arbitrary symmetric ideals. Moreover, the slope of the linear function is explicitly determined. We conjecture that the projective dimensions also grow eventually linearly. As part of the evidence we establish two non-trivial lower linear bounds of the projective dimensions for chains of monomial ideals. As an application, this yields Cohen-Macaulayness obstructions. K E Y W O R D S invariant ideal, monoid, polynomial ring, symmetric group M S C ( 2 0 1 0 ) 13A50, 13C15, 13D02, 13F20, 16P70, 16W22 Let Sym( ) denote the symmetric group on {1, … , }. Considering it as stabilizer of + 1 in Sym( + 1), similarly one gets an ascending chain of symmetric groups. Define an action of Sym( ) on induced by ⋅ , = , ( ) for every ∈ Sym( ), 1 ≤ ≤ , 1 ≤ ≤ . 346

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Finite generation of symmetric ideals

Cited by 79 publications

References 17 publications

Minimal generators for symmetric ideals

Minimal generators for symmetric ideals

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

Codimension and projective dimension up to symmetry

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