Minimal generators for symmetric ideals

Abstract

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

Noetherianity and Specht problem for varieties of bicommutative algebras

“…Again, as in Remarks 3.3 and 4.2 we can ask about the number of generators of the T-ideals of F (B) when the base field is of positive characteristic. Since we work in the polynomial algebra K[Y, Z] considered as a K[X]-bimodule we shall mention several results concerning the number of generators, theory of Gröbner bases and other algorithmic problems: Aschenbrenner, Hillar [2], Hillar, Windfeldt [20], Hillar, Sullivant [19], Krone [24], and Hillar, Krone, Leykin [18].…”

An algorithm for finding symmetric Grobner bases in infinite dimensional rings

“…Symmetric ideals can be arbitrarily complex in the following sense. For each n, there are symmetric ideals of R that cannot have fewer than n R[S ∞ ]module generators [5]. Moreover, such ideals are not always monomial.…”