The diagonal fallacy is my name for the mistaken belief that the direct sum decomposition of as the direct sum is in some way unique. It is frequently committed by undergraduates, many of whom have the even more mistaken belief that somehow means . (Probably they are being mislead by their deep insight that and are both categorical coproducts.)
James’ Submodule Theorem is a fundamental theorem in the representation theory of the symmetric group. To give an illustrative example, fix a prime , and let be the permutation module for acting on the set of all ordered pairs with and . Thus the elements of are formal linear combination of the elements of with coefficients in the finite field . Let be the bilinear form on defined on basis elements so that
Write for the orthogonal complement of a subspace of with respect to this form. Let
The Specht module can be defined as the -submodule of generated by the element
Writing for , a special case of James’ Submodule Theorem states that if is a -submodule of then either , in which case , or , in which case it follows from the definition of that .
A corollary is that in any direct sum decomposition of there is a unique indecomposable summand containing , namely the unique summand such that . This summand is unique up to isomorphism, by the Krull–Schmidt theorem.
If is odd then is in fact unique as a submodule of , since then is a summand of the submodule of , and this module is indecomposable if divides and otherwise semisimple with two non-isomorphic summands (so no `diagonal' submodules), namely the Specht modules and . When the situation is more complicated, and I think it would be interesting to know whether is unique as a submodule of .
The following example shows one way uniqueness can fail.
Example. Let be a field and let . Then has a two-dimensional module on which acts by , , and a one-dimensional module on which acts by . There are distinct direct sum decompositions of , both with a unique summand containing , namely
Note that , but evidently there is no ‘unique submodule theorem’ applicable to the image of the map .