Betti numbers of symmetric shifted ideals

TL;DR

This paper introduces symmetric shifted ideals, a new class of monomial ideals fixed by the symmetric group, and computes their Betti numbers, with applications to symbolic powers of star configuration ideals.

Contribution

The paper defines symmetric shifted ideals, proves they have linear quotients, and calculates their equivariant graded Betti numbers, extending understanding of symmetric monomial ideals.

Findings

Symmetric shifted ideals have linear quotients.

Explicit formulas for equivariant graded Betti numbers are derived.

Applications to symbolic powers of star configuration ideals are demonstrated.

Abstract

We introduce a new class of monomial ideals which we call symmetric shifted ideals. Symmetric shifted ideals are fixed by the natural action of the symmetric group and, within the class of monomial ideals fixed by this action, they can be considered as an analogue of stable monomial ideals within the class of monomial ideals. We show that a symmetric shifted ideal has linear quotients and compute its (equivariant) graded Betti numbers. As an application of this result, we obtain several consequences for graded Betti numbers of symbolic powers of defining ideals of star configurations.