Equivariant algebraic properties of monomial ideals under the Inci(ℕ) operator

In this paper, we introduce the concept of equivariant linear quotients in [Formula: see text], extending the notion of linear quotients for monomial ideals in the finite polynomial ring [Formula: see text] over a field [Formula: see text]. We also define the notions of equivariant stability and equivariant strong stability. Our main results show that if [Formula: see text] is a square-free monomial ideal with equivariant linear quotients, then the ideal [Formula: see text] also possesses equivariant linear quotients (see Theorem 3.12). We demonstrate that if [Formula: see text] is a square-free equivariant stable ideal, then [Formula: see text] remains a square-free equivariant stable ideal (see Theorem 4.7). Moreover, we provide examples illustrating that these results do not generally extend to non-squarefree ideals. Alongside, we examine the asymptotic behavior of the Castelnuovo–Mumford regularity and the projective dimension of these chains of ideals.