Group Actions and Representations
From the module perspective, we’re led back to the concept of a group action. This is like a -module, but “discrete”. Let’s just write down the axioms for easy reference: we’ve got a set and a function such that
- preserves the identity: .
- preserves the group operation: .
Notice how this looks almost exactly like the axioms for a -module, except since is just a set we don’t have any sense of “linearity”.
Now, from a group action on a finite set we can get a finite-dimensional representation. We let — the vector space defined to have as a basis. That is, vectors in are of the form
for some complex coefficients . We get a -module by extending to a bilinear function . We already know how it behaves on the basis of the form , and the extension to a bilinear function is uniquely defined. We call the “permutation representation” associated to , and the elements for we call the “standard basis”.
As an example, the group is defined from the very beginning by the fact that it acts on the set by shuffling the numbers around. And so we get a representation from this action, which we call the “defining representation”. By definition, it has dimension , since it has a basis given by . To be even more explicit, let me write out the defining matrix representation for . Technically, going from an abstract representation to a matrix representation requires not just a basis, but an ordered basis, but the order should be pretty clear in this case. And so, with no further ado:
To see how this works, note that the permutation sends to . Similarly, we find that . That is:
We also see that composition of permutations turns into matrix multiplication. For example, . In terms of the matrices we calculate:
You can check for yourself all the other cases that you care to.
Notice that in general the matrices are index by two elements of , and the matrix element — the one in the th row and th column — is . That is, it’s if — if the action of sends to — and otherwise. This guarantees that every entry will be either or , and that each row and each column will have exactly one . Such a matrix we call a “permutation matrix”, and we see that the matrices that occur in permutation representations are permutation matrices.