Wolfforms

Wolfforms is a term that appears in several niche mathematical writings, but there is no single, widely accepted definition. In general, wolfforms are described as a class of homogeneous polynomial forms on a finite-dimensional vector space that are defined with respect to a symmetry or weight structure. A common reading treats a wolfform of degree d on a vector space V over a field F as a multilinear form that can be represented by a symmetric d-tensor and that satisfies an invariance property under a prescribed group G of linear transformations, namely f(g·v) = f(v) for all g in G and v in V. The precise choice of G or the weighting used to define the form can vary between sources, leading to several related notions.

Two typical variants appear in the literature. The first is invariant wolfforms: homogeneous forms f of degree d that are invariant under a given group action. The second is weighted or generalized wolfforms: forms defined by a coefficient tensor together with a weight function that modifies the invariance condition.

Examples include quadratic wolfforms, which in two dimensions invariant under a subgroup of GL(2) reduce to standard invariant quadratic forms; higher-degree wolfforms generalize these to cubic or higher polynomials with similar symmetry constraints.

In applications, wolfforms commonly arise in invariant theory, representation theory, algebraic geometry, and computer algebra, where they help classify symmetries of polynomial expressions or construct objects with prescribed invariance properties.

Because the term is not standardized, readers should consult the specific source for the exact definition used in that context. See also invariant form, multilinear form, polynomial form, tensor.