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

Abstract

Let K be a field and let ℕ = {1, 2, . . .} be the set of all positive integers. Let R n = K[x ij | 1 ≤ in, j ∈ ℕ] be the ring of polynomials in x ij (1 ≤ in, j ∈ ℕ) over K. Let S n = Sym({1, 2, . . . , n}) and Sym(ℕ) be the groups of permutations of the sets {1, 2, . . . , n} and ℕ, respectively. Then S n and Sym(ℕ) act on R n in a natural way: τ (x ij ) = x τ(i)j and σ(x ij ) = x (j), for all i ∈ {1, 2, . . . , n} and j ∈ ℕ, τ ∈ S n and σ ∈ Sym(ℕ). Let \( \overline{R} \) n be the subalgebra of (S n -)symmetric polynomials in R n , i.e.,

$$ {\overline{R}}_n=\left\{f\in {R}_n\left|\tau (f)=f\;\mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{each}\ \tau \right.\in {\mathrm{S}}_n\right\}. $$

An ideal I in \( \overline{R} \) n is called Sym(ℕ)-invariant if σ(I) = I for each σ ∈ Sym(ℕ). In 1992, the second author proved that if char(K) = 0 or char(K) = p > n, then every Sym(ℕ)-invariant ideal in \( \overline{R} \) n is finitely generated (as such). In this note, we prove that this is not the case if char(K) = pn. We also survey some results on Sym(ℕ)-invariant ideals in polynomial algebras and some related topics.

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References

  1. 1.

    M. Aschenbrenner and C. J. Hillar, “Finite generation of symmetric ideals,” Trans. Am. Math. Soc., 359, 5171–5192 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    A. E. Brouwer and J. Draisma, “Equivariant Gröbner bases and the Gaussian two-factor model,” Math. Comput., 80, 1123–1133 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    D. E. Cohen, “On the laws of a metabelian variety,” J. Algebra, 5, 267–273 (1967).

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    D. E. Cohen, “Closure relations, Buchberger’s algorithm, and polynomials in infinitely many variables,” in: Computation Theory and Logic, Lect. Notes Comput. Sci., Vol. 270, Springer, Berlin (1987), pp. 78–87.

  5. 5.

    G. S. Deryabina and A. N. Krasilnikov, “On solvable groups of exponent 4,” Sib. Math. J., 44, 58–60 (2003).

    Article  MathSciNet  Google Scholar 

  6. 6.

    J. Draisma, “Finiteness for the k-factor model and chirality varieties,” Adv. Math., 223, 243–256 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

    J. Draisma, Noetherianity up to Symmetry, arXiv:1310.1705 (2013).

  8. 8.

    J. Draisma, R. H. Eggermont, R. Krone, and A. Leykin, Noetherianity for Infinite-Dimensional Toric Varieties, arXiv:1306.0828 (2013).

  9. 9.

    J. Draisma and A. Krasilnikov, “Sym(ℕ)-invariant ideals in a certain subalgebra of a polynomial algebra,” in preparation.

  10. 10.

    J. Draisma and J. Kuttler, “On the ideals of equivariant tree models,” Math. Ann., 344, 619–644 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  11. 11.

    J. Draisma and J. Kuttler, “Bounded-rank tensors are defined in bounded degree,” Duke Math. J., 163, No. 1, 1–266 (2014).

    Article  MathSciNet  Google Scholar 

  12. 12.

    C. J. Hillar and A. Martín del Campo, “Finiteness theorems and algorithms for permutation invariant chains of Laurent lattice ideals,” J. Symbol. Comput., 50, 314–334 (2013).

    Article  MATH  Google Scholar 

  13. 13.

    C. J. Hillar and S. Sullivant, “Finite Gröbner bases in infinite dimensional polynomial rings and applications,” Adv. Math., 229, 1–25 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  14. 14.

    A. N. Krasilnikov, “Identities of triangulable matrix representations of groups,” Trans. Moscow Math. Soc., 1990, 233–249 (1991).

    MathSciNet  Google Scholar 

  15. 15.

    A. N. Krasilnikov, “The identities of a group with nilpotent commutator subgroup are finitely based,” Math. USSR Izv., 37, 539–553 (1991).

    Article  MathSciNet  Google Scholar 

  16. 16.

    A. N. Krasilnikov, “Identities of Lie algebras with nilpotent commutator ideal over a field of finite characteristic,” Math. Notes, 51, 255–258 (1992).

    Article  MathSciNet  Google Scholar 

  17. 17.

    M. R. Vaughan-Lee, “Abelian-by-nilpotent varieties of Lie algebras,” J. London Math. Soc. (2), 11, 263–265 (1975).

  18. 18.

    M. R. Vaughan-Lee, “Abelian by nilpotent varieties,” Quart. J. Math. Oxford Ser. (2), 21, 193–202 l(1970).

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Correspondence to E. A. da Costa.

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Dedicated to Alfred Shmelkin on the occasion of his 75th birthday and to Viktor Markov on the occasion of his 65th birthday

Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 18, No. 3, pp. 69–76, 2013.

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da Costa, E.A., Krasilnikov, A. Symmetric Polynomials and Nonfinitely Generated Sym(ℕ)-Invariant Ideals. J Math Sci 206, 505–510 (2015). https://doi.org/10.1007/s10958-015-2329-1

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Keywords

  • Polynomial Algebra
  • Symmetric Polynomial
  • Metabelian Group
  • Invariant Ideal
  • Nilpotent Variety