Characterization of equivariantly multiplicative stably Pfister forms

Determine to what extent stably Pfister quadratic forms equipped with a finite-group action are equivariantly multiplicative, thereby characterizing when the stable Pfister property admits an equivariant multiplicative realization.

Background

The paper defines a non-degenerate finite-group-invariant quadratic form to be equivariantly multiplicative when there is a finite-group-equivariant rational map from the underlying vector space to its similitude group satisfying the multiplicative identity q(τ(x)·y)=q(x)q(y). The authors prove that every equivariantly multiplicative form is stably Pfister; in particular, its dimension is a power of two and its discriminant is trivial in dimensions greater than two.

The converse is not established. The question asks whether, and under what additional hypotheses, the stable Pfister condition can be upgraded to the existence of an equivariant rational multiplicative law. This is included because the paper’s stable theory is defined through twists and rational maps to maximal isotropic Grassmannians, whereas equivariant multiplicativity is a stronger unstable structure.