Symmetric Polynomials and Nonfinitely Generated Sym(ℕ)-Invariant Ideals
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| 024 | 7 | 0 | |a 10.1007/s10958-015-2329-1 |2 doi |
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| 245 | 0 | 0 | |a Symmetric Polynomials and Nonfinitely Generated Sym(ℕ)-Invariant Ideals |h [Elektronische Daten] |c [E. da Costa, A. Krasilnikov] |
| 520 | 3 | |a Let K be a field and let ℕ = {1, 2, . . .} be the set of all positive integers. Let R n = K[x ij | 1 ≤ i ≤ n, j ∈ ℕ] be the ring of polynomials in x ij (1 ≤ i ≤ n, 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 iσ(j), for all i ∈ {1, 2, . . . , n} and j ∈ ℕ, τ ∈ S n and σ ∈ Sym(ℕ). Let R ¯ $$ \overline{R} $$ n be the subalgebra of (S n -)symmetric polynomials in R n , i.e., R ¯ n = f ∈ R n τ f = f f o r each τ ∈ S n . $$ {\overline{R}}_n=\left\{f\in {R}_n\left|\tau (f)=f\;\mathrm{f}\mathrm{o}\mathrm{r}\ \mathrm{each}\ \tau \right.\in {\mathrm{S}}_n\right\}. $$ An ideal I in R ¯ $$ \overline{R} $$ 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 R ¯ $$ \overline{R} $$ n 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 on Sym(ℕ)-invariant ideals in polynomial algebras and some related topics. | |
| 540 | |a Springer Science+Business Media New York, 2015 | ||
| 700 | 1 | |a da Costa |D E. |u Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil |4 aut | |
| 700 | 1 | |a Krasilnikov |D A. |u Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil |4 aut | |
| 773 | 0 | |t Journal of Mathematical Sciences |d Springer US; http://www.springer-ny.com |g 206/5(2015-05-01), 505-510 |x 1072-3374 |q 206:5<505 |1 2015 |2 206 |o 10958 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s10958-015-2329-1 |q text/html |z Onlinezugriff via DOI |
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| 949 | |B NATIONALLICENCE |F NATIONALLICENCE |b NL-springer | ||
| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s10958-015-2329-1 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a da Costa |D E. |u Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil |4 aut | ||
| 950 | |B NATIONALLICENCE |P 700 |E 1- |a Krasilnikov |D A. |u Departamento de Matemática, Universidade de Brasília, 70910-900, Brasília, DF, Brazil |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Journal of Mathematical Sciences |d Springer US; http://www.springer-ny.com |g 206/5(2015-05-01), 505-510 |x 1072-3374 |q 206:5<505 |1 2015 |2 206 |o 10958 | ||