singletensors

A singletensor is a tensor that belongs to the trivial (singlet) representation of a symmetry group. In more concrete terms, if a group G acts on a vector space V and on any tensor product built from V, a tensor T is a singletensor when applying any group element leaves T unchanged: g · T = T for all g in G. Such tensors are invariant under the group action and are used to build quantities that do not transform under the symmetry.

The space of singletensors is the G-invariant subspace of the tensor product in question. For compact groups, a standard way to obtain a singletensor is to average over the group: T_inv = ∫ g·T dg, where the integral runs over the group with respect to the Haar measure. Finite groups allow a similar averaging by summing over all group elements. In practice, singletensors often arise by contracting indices with invariant tensors of the group, producing scalars or objects that transform trivially.

Examples include:

- For the orthogonal or special orthogonal groups, the Kronecker delta δ_ij is an invariant tensor that can be used to form invariants by contracting indices.

- For SU(2), the antisymmetric symbol ε_ab serves as an invariant tensor that can produce singlet combinations from spinor representations (e.g., combining two spin-1/2 states to yield a total spin-0 state).

- For SU(3) color symmetry in particle physics, invariant tensors like δ_ij or ε_ijk are used to form color-singlet combinations of quark fields.

Applications span constructing invariant Lagrangians, classifying singlet states in quantum systems, and analyzing how composite objects couple to preserve symmetry. Note that the term singletensor is not universally standard; more common phrasing is “invariant tensor” or “tensor transforming in the singlet representation.”