14.23.3 Tensor Map Product Operator Representation Boundary

Tensor Map Product Operator Representation Boundary is the delineation of where the specific results established for operators, namely the eigenvalue product formula and the spectral behavior of the tensor product of operators, and for representations, namely the character multiplicativity and the decomposition into irreducibles, cease to hold in their stated form once the operators fail to be diagonalizable or the representations fail to be completely reducible.

Boundary at Non-Diagonalizable Operators

The Eigenvalue Formula Still Applies

The statement that every eigenvalue of $T\otimes S$ is a product $\lambda \mu$ of an eigenvalue of $T$ and an eigenvalue of $S$ continues to hold even when $T$ or $S$ is not diagonalizable, since the eigenvalue relation is established directly from the action of $T\otimes S$ on a genuine eigenvector $v\otimes w$, which exists whenever $T$ and $S$ individually have at least one eigenvector each.

Where the Diagonalization Result Fails

The stronger claim, that $T\otimes S$ is diagonalizable whenever $T$ and $S$ are, has no converse or extension once either factor is merely triangularizable rather than diagonalizable; if $T$ has a nontrivial Jordan block, the Jordan structure of $T\otimes S$ is generally more intricate than a simple product of the two factors' Jordan structures, and no closed formula analogous to the diagonalizable case applies at this boundary; determining the exact Jordan form of a Kronecker product of non-diagonalizable matrices is a separate computational problem not solved by the tensor map product construction alone.

Boundary at Non-Semisimple Representations

Complete Reducibility Requires Extra Hypotheses

The claim that a tensor product representation $\rho \otimes \sigma$ of irreducible representations decomposes as a direct sum of irreducibles relies on complete reducibility of the representation category, which for finite groups requires, by Maschke's theorem, that the characteristic of the field not divide the order of the group; outside this hypothesis, in modular representation theory, a tensor product of irreducible representations may fail to decompose into a direct sum at all, instead admitting only a composition series with repeated or non-split extensions between its factors.

Consequence for the Character Formula

The character multiplicativity formula ${\chi }_{\rho \otimes \sigma }\left(g\right)={\chi }_{\rho }\left(g\right)\cdot {\chi }_{\sigma }\left(g\right)$ itself continues to hold regardless of complete reducibility, since it is a direct consequence of the trace behavior of the Kronecker product and requires no decomposition; what fails at this boundary is only the further step of reading off multiplicities of irreducible constituents from the character via orthogonality relations, since those orthogonality relations themselves depend on complete reducibility.

Boundary at Infinite Groups

Loss of a Finite Character Table

For a finite group, the decomposition of a tensor product representation into irreducibles is governed by a finite character table and finite orthogonality sums; for an infinite group, no such finite table exists in general, and the analogous decomposition, when it exists at all, requires either a topology on the group, as for Lie groups and their continuous representations, or a substitute notion of orthogonality using an invariant measure, as in the Peter–Weyl theory for compact groups.

Where Even This Weaker Structure Fails

For a general infinite, non-compact group with no invariant probability measure available, tensor product representations may fail to decompose into irreducibles in any useful sense at all, and the representation role of the tensor product of maps, while still producing a well-defined representation on the tensor product space, no longer comes with any of the decomposition machinery developed for the finite or compact cases.

Boundary at Infinite-Dimensional Operator Algebras

Continuity and Domain Issues

When $T$ and $S$ are bounded operators on infinite-dimensional Hilbert spaces, the tensor product operator $T\otimes S$ extends continuously to the completed tensor product space, and the eigenvalue product statement continues to hold for eigenvectors, when they exist; for unbounded operators, however, the domain of $T\otimes S$ becomes a delicate issue, since a dense domain for each factor does not automatically produce a well-behaved dense domain for the combined operator, marking a boundary where the purely algebraic construction of the tensor product of maps must be supplemented with analytic domain considerations absent from the finite-dimensional theory.