Test, multiplier and invariant ideals

TL;DR

This paper provides explicit formulas for multiplier and test ideals of invariant ideals in symmetric algebras, linking algebraic invariants to combinatorial polytopes derived from Young diagrams.

Contribution

It introduces a unified approach to compute multiplier and test ideals for a broad class of invariant ideals using geometric and combinatorial methods.

Findings

Explicit formulas for multiplier ideals of GL(V)xGL(W)-invariant ideals.

Computed test ideals and F-pure thresholds for sums of determinantal ideals.

Recovered known formulas for monomial ideals and extended to new classes of invariant ideals.

Abstract

This paper gives an explicit formula for the multiplier ideals, and consequently for the log canonical thresholds, of any GL(V)xGL(W)-invariant ideal in the symmetric algebra S of the tensor product of V with the dual of W, where V and W are vector spaces over a field of characteristic 0. This characterization is done in terms of a polytope constructed from the set of Young diagrams corresponding to the Schur modules generating the ideal. Our approach consists in computing the test ideals of some invariant ideals of S in positive characteristic: Namely, we will compute the test ideals (and so the F-pure thresholds) of any sum of products of determinantal ideals. Even in characteristic 0, not all the invariant ideals are as the latter, but they are up to integral closure, and this is enough to reach our goals. The results concerning the test ideals are obtained as a consequence of general results holding true in a special situation. Within such framework fall determinantal objects of a generic matrix, as well as of a symmetric matrix and of a skew-symmetric one. Similar results are thus deduced for the GL(V)-invariant ideals in the symmetric algebra of the second symmetric (or alternating) power of V. (Also monomial ideals fall in this framework, thus we recover Howald's formula for their multiplier ideals and, more generally, we get the formula for their test ideals). During the proof, we introduce the notion of "floating test ideals", a property that in a sense is satisfied by ideals defining schemes with singularities as nice as possible. As we will see, products of determinantal ideals, and by passing to characteristic 0 ideals generated by a single Schur module, have this property.