14.22.3 Tensor Map Product Operator Notation

The Tensor Map Product Operator Notation formalizes how tensors interact under mappings, providing a structured way to compose tensor operations in algebraic contexts.

Tensor Map Product Operator Notation is the set of notational shortcuts used specifically when the tensor product of maps is applied to endomorphisms, allowing an operator acting on only one tensor factor to be written without explicitly displaying the identity operator on the other factor, and allowing subscripted labels to indicate which factor an operator acts on within a fixed combined space.

Suppressing the Identity Factor

The Full Expression

An operator acting nontrivially only on $V$ , leaving $W$ untouched, is properly written $T \otimes {id}_{W}$ , an explicit tensor product of $T$ with the identity map on $W$ , as required by the general tensor product of maps construction.

The Abbreviated Form

Once the combined space $V \otimes W$ has been fixed in a discussion, it is common to abbreviate $T \otimes {id}_{W}$ simply as $T$ , relying on context to indicate that $T$ is being regarded as an operator on the larger space $V \otimes W$ rather than on $V$ alone; this abbreviation is common in physics-oriented treatments where writing out the identity factor on every term would clutter otherwise simple expressions.

Subscript Notation for Factor Labels

Numbering the Tensor Factors

When several copies of the same or similar spaces are tensored together, for instance $V \otimes V \otimes V$ , an operator acting only on the first copy is written $T_{1}$ , meaning $T \otimes id \otimes id$ , an operator acting only on the second copy is written $T_{2}$ , meaning $id \otimes T \otimes id$ , and so on, with the subscript indicating the position of the nontrivial factor among an otherwise implicit sequence of identity operators.

Consistency With Commutativity

Because operators labeled with different subscripts, such as $T_{1}$ and $S_{2}$ , always commute, this notation makes the commuting operator construction described elsewhere immediately visible from the distinct subscripts alone, without needing to invoke the interchange law explicitly each time.

Notation for Sums of Single-Factor Operators

Additive Combinations Across Factors

A very common construction, particularly for operators meant to describe a total quantity distributed across a combined system, is written as a sum of single-factor operators,

T \otimes id + id \otimes S

abbreviated $T_{1} + S_{2}$ under the subscript convention, in contrast with the pure tensor product $T \otimes S$ , which acts jointly on both factors at once rather than additively on each factor separately.

Distinguishing Additive From Multiplicative Combinations Notationally

The subscript sum notation makes visually explicit a distinction that is easy to blur without it: $T_{1} + S_{2}$ and $T \otimes S$ are generally very different operators on $V \otimes W$ , and writing one out fully as a sum of two Kronecker products while the other remains a single Kronecker product keeps this difference visible even after the identity factors have been suppressed elsewhere.

Notation for the Diagonal Case in Representation Theory

Same Operator on Every Factor

When a single group element $g$ acts identically in spirit on every tensor factor of a representation, as in the tensor product representation role, the notation $ρ (g) \otimes σ (g)$ is used directly, with the shared argument $g$ distinguishing this from the operator construction role's notation, where two independent elements from possibly different groups would instead appear.

Bracketed Operator Notation

In some conventions, particularly in more physics-oriented sources, an operator understood to act on a full tensor product space is enclosed in square or angle brackets alongside the space it acts on, such as $[T \otimes S] on V \otimes W$ , purely as a typographical aid to keep the domain of discourse explicit in a longer derivation, rather than as a mathematically distinct symbol.