Multisymmetric polynomials in dimension three

Abstract

The polarizations of one relation of degree five and two relations of degree six minimally generate the ideal of relations among a minimal generating system of the algebra of multisymmetric polynomials in an arbitrary number of three-dimensional vector variables. In the general case of n-dimensional vector variables, a relation of degree 2n among the polarized power sums is presented such that it is not contained in the ideal generated by lower degree relations.MSC: 13A50, 14L30, 20G05

Applications of Multisymmetric Syzygies in Invariant Theory

“…(ii) Although the presentation of T n (A) S n given in Theorem 1 is infinite, in certain cases an a priori upper bound for the degrees of relations in a minimal presentation is available, and a finite presentation can be obtained from the infinite presentation above (see for instance [10, Theorem 3.2], building on [8]). Based on this procedure even a minimal presentation is worked out in [11] for…”

“…Denote by φ ′ the restriction of φ to the nine-variable polynomial algebra It was explained first in [10] how to deduce from a special case of Theorem 1 a minimal generating system of ker(µ). Later in [11] a natural action of the general linear group GL 2 (K) was taken into account and it was proved that ker(µ) is minimally generated as a GL 2 (K)-ideal by J 3,2 and J 4,2 given as follows:…”

“…In Section 5 we turn to a very concrete application of Theorem 1. Its special case when A = k[x 1 , x 2 ] is a two-variable polynomial ring was used in [10], [11] to derive generators of the ideal of relations among a minimal generating system of T 3 (A) S 3 . This is applied here to give a minimal presentation of the ring of invariants R S 4 of the permutation representation of the symmetric group S 4 associated with the action of S 4 on the set of two-element subsets of the set {1, 2, 3, 4}.…”

“…, x 2 , x 3 , y 1 , y 2 , y 3 ] similarly to the identification of T 3 (K[x, z]) and R). Therefore we have φ ′ =ψ • µ(9) where µ stands for the K-algebra surjectionµ : F ′ → T 3 (K[x, y]) S 3 , T w → [w] for w ∈ {x, x 2 , x 3 , y, y 2 , y 3 ,xy, x 2 y, xy 2 } studied in[10] and[11]. In particular, since T 3 (K[x, y]) S 3 is known to be minimally generated by the elements [w] with w ∈ {x, x 2 , x 3 , y, y 2 , y 3 , xy, x 2 y, xy 2 }, the K-algebra homomorphisms µ and hence φ ′ are surjective onto T 3 (K[x, y]) S 3 , respectively onto P S 3 .…”

Separating invariants for multisymmetric polynomials

“…These sets also show that the upper bound (1.5) is exact for n ≤ 4. For n = 3 and K = C the algebra K[V m ] S3 was studied in details by Domokos and Puskás [7]. This paper is structured as follows.…”