Let be a finite group and let be a representation of by matrices with rational entries. We regard these matrices as acting on row vectors in an -dimensional vector space with standard basis . The -submodule of spanned by is free of rank , and so has a -basis, say . In this new basis, the matrices representing elements of have integer entries. We may therefore assume that has image in .
It is now easy to obtain a representation of in prime characteristic : just think of the entries of each as lying in rather than . A representation obtained in this way is called a -modular reduction of . The indefinite article is correct: -modular reductions defined with respect to different -bases need not be isomorphic. However, as in the Jordan–Hölder Theorem, they always have the same multiset of composition factors. (For a proof, see Theorem 32 in Serre, Linear representations of finite groups.)
I think of representations of finite groups as rather ‘rigid’ objects, so find it slightly surprising how much freedom there often is to obtain different -modular reductions. Here is an example from the symmetric group. Switching to the language of modules, let , let be a prime dividing , and let be the natural permutation module for . The subspace
of is a -submodule. One obvious -basis for is . Provided , in this basis, the matrices representing the generators and of are
respectively. Consider the -basis for in which is replaced with
Since the coefficient of in is , the change of basis matrix is triangular with s on its diagonal. Therefore is a -basis for . In this basis the matrices representing the chosen generators are conjugated to
Let be the -module obtained by -modular reduction. Since divides , we see from the final row of the matrices immediately above that is a -dimensional trivial -submodule of . The quotient is a simple -module, say. By the result on composition factors mentioned earlier, any -modular reduction of has composition factors and .
Suppose we scale the final basis vector by . If the coefficient of in is , then the coefficient of in is . Similarly, if the coefficient of in is then the coefficient of in is . The effect on matrices is to scale the final column by , and the final row by , leaving the entry in the bottom right unchanged. The new matrices are therefore
respectively. Let be the highest power of dividing . Taking we see that has a -modular reduction in which appears as the unique bottom composition factor, and as the unique top composition factor. Moreover, if then by taking we get a semisimple -modular reduction of .
This trick for moving composition factors up or down in the radical series of a module generalizes. In fact, over a suitable ring, one may move composition factors around almost at will. One striking application is Lemma 18.2 in Feit, Finite linear groups. The special case for rational representations is proved below.
Theorem. Let be a finite group and let be a -module. There is a field containing such that the -module has a semisimple -modular reduction.
Proof. Suppose that has a -modular reduction having exactly composition factors. There is an -basis of in which the representing matrices have the block form
Let . Let . Scaling the basis vectors corresponding to the row containing by for each gives a -basis of in which the representing matrices are
The corresponding -modular reduction, defined by quotienting by its maximal ideal , is semisimple.
The following theorem gives another generalization of the example above. Given a commutative ring and a partition , let denote the corresponding Specht module for .
Theorem. Let be a partition. There is a -modular reduction of isomorphic to .
Proof. Let . Let be a tableau of shape with distinct entries from . Let , respectively , where the sums are over all permutations that permute amongst themselves the entries of each row, respectively column, of . There are isomorphisms and . Moreover, is a pseudo-idempotent, i.e. for some non-zero . (For proofs of these facts see Lemma 5 and Exercise 20 in Chapter 7 of Fulton, Young tableaux.) Hence defines an injective homomorphism . Therefore has a -basis defining a reduction modulo of isomorphic to .
An alternative description of the inclusion homomorphism in this proof is given by Fayers in Proposition 4.7 of On the structure of Specht modules. The results on Schaper layers in his paper can be used to give many further examples of semisimple -modular reductions of Specht modules that do not require a field extension.