**Definition**. Two representations are called **equivalent** if as -modules.

**Remark 1**. If as -modules, then obviously as -modules, i.e. The converse is not necessarily true though. Try to find a counter-example!

**Remark 2**. Let’s see what exactly “equivalent representations” means. Suppose that

are equivalent. So there exists a -module isomorphism Thus for all we have

But, by definition and because Hence

and so Therefore, since is an isomorphism, we get that for all

**Remark 3**. It follows from that the degree one representations are equivalent iff they are equal. This is clear because, in terms of matrices, and are matrices in this case and so scalars. Thus gives us for all

Advertisements