14.13.3 Tensor Map Product Parenthesis Suppression

Tensor Map Product Parenthesis Suppression is the notational convention of omitting parentheses entirely from a tensor product of three or more maps, justified by the fact that every possible placement of parentheses among the factors produces the same combined operator, so no ambiguity arises from leaving the grouping unspecified.

Justification for Suppressing Parentheses

Associativity as the Enabling Property

Because the tensor product of maps is associative, any two ways of inserting parentheses among a fixed sequence of factors yield the same final combined operator, which removes the need to specify a grouping at all when writing the expression.

The Suppressed Form

Once this agreement is established, the expression is written with all internal parentheses removed, leaving only the sequence of factors joined by the tensor product symbol.

What Is Preserved After Suppression

Order of the Factors Remains Essential

Suppressing the parentheses removes only the information about grouping; the left-to-right order in which the factors are listed is still essential and must be preserved exactly, since reordering the factors is governed by entirely separate considerations from grouping.

The Underlying Tensor Product Space Is Also Written Without Parentheses

Consistent with suppressing parentheses in the operator expression, the tensor product of the underlying vector spaces is written the same way, without parentheses, relying on the same associativity property at the level of spaces rather than maps.

Diagram Contrasting Explicit and Suppressed Notation

Same Operator, Simplified Notation

The diagram below contrasts an explicitly parenthesized expression with its suppressed form, both representing the identical combined operator.

When Suppression Is Not Appropriate

Mixed Expressions Involving Composition

When an expression mixes tensor products of maps with ordinary composition of operators, parentheses generally cannot be suppressed freely, since composition is not associative with the tensor product in every possible grouping without careful attention to which operators act on which spaces.

Emphasizing an Intermediate Grouping for Computation

Even though any grouping is mathematically equivalent, a specific grouping is sometimes reintroduced deliberately in a written expression to highlight which pair of factors is being combined first for computational or expository purposes, temporarily reversing the suppression for clarity.

Suppression at the Matrix Level

Kronecker Product Chains Without Parentheses

The same suppression convention applies to chains of Kronecker products of matrices, where associativity of the Kronecker product justifies writing a long product of factor matrices without any internal parentheses.

Implementation Detail Hidden by the Notation

Although the notation suppresses the grouping, an actual numerical computation of a suppressed expression still must choose some specific order in which to carry out the individual pairwise Kronecker products, with that choice affecting only the intermediate computational steps and not the final composite matrix.

Suppression Combined With Identity and Reassociation

Compatibility With Identity Simplifications

Suppressed notation does not prevent recognizing and simplifying identity factors within the sequence, since an identity factor can still be identified by its position in the unparenthesized list and simplified according to the identity map composition compatibility already established.

Suppression as the Endpoint of Reassociation

Parenthesis suppression can be viewed as the natural endpoint of applying the reassociation rule repeatedly: once every grouping is known to agree, there is no remaining reason to track any particular arrangement of parentheses, and the fully suppressed form becomes the standard way of writing the expression.