tensorinen

Tensorinen is a term encountered in certain mathematical discussions as a proposed generalization of tensors. It describes a class of multilinear objects that carry an additional discrete grading, called the tensorinen degree, alongside their traditional tensor indices. Formally, for a finite-dimensional vector space V over a field F and a fixed abelian grading group G, a tensorinen of type (p,q) consists of a multilinear map T: V*^p × V^q → F whose components are labeled not only by the usual indices i1,...,ip, j1,...,jq but also by a degree e ∈ G. Under a change of basis, the components transform like ordinary tensors and additionally carry a weight determined by e.

Operations on tensorinen mirror those of ordinary tensors, including addition, scalar multiplication, and contraction. One can define a tensorinen product T ⊗ S that combines degrees via deg(T ⊗ S) = deg(T) + deg(S). When the grading is trivial (G is the zero group), tensorinen reduces to the standard tensor of type (p,q). Contraction is permitted only between a covariant and contravariant index in a way that preserves the tensorinen degree.

In practice, tensorinen has been proposed as a framework to encode discrete symmetries or parity information alongside geometric data. It is mainly discussed in theoretical expositions and is not part of the standard toolkit of differential geometry or representation theory. As such, tensorinen serves as a conceptual extension rather than a widely used construction, with concrete applications remaining an area of speculative research.