A useful construction for our purposes is the group algebra . We’ve said a lot about this before, and showed a number of things about it, but most of those we can ignore for now. All that’s really important is that is an algebra whose representations are intimately connected with those of .
When we say it’s an algebra, we just mean that it’s a vector space with a distributive multiplication defined on it. And in our case of a finite group it’s easy to define both. For every group element we have a basis vector . That is, we get every vector in the algebra by picking one complex coefficient for each element , and adding them all up:
Multiplication is exactly what we might expect: the product of two basis vectors and is the basis vector , and we extend everything else by linearity!
We often rearrange this sum to collect all the terms with a given basis vector together:
And we can go from representations of a group to representations of its group algebra. Indeed, if is a representation, then we can define
Indeed, each is a matrix, and we can multiply matrices by complex numbers and add them together, so the right hand side is perfectly well-defined as a matrix in the matrix algebra . It’s a simple matter to verify that preserves addition of vectors, scalar multiples of vectors, and products of vectors.
Conversely, if is a representation, then we can restrict it to our basis vectors and get a map . The image of each basis vector must be invertible, for we have