I want to show that the representation above is actually a representation and is also irreducible. Since (12) and (123) are generators for S3, is it correct to say that all I need to do is to show that the order of $p(12)$ is the same as (12) and the order of $p(123)$ is the same as (123) as well as the following homomorphism: $p(12)(123)=p(12)p(123)$ and $p(123)(12)=p(123)p(12)$? I am not sure if this is sufficient and what the criteria are for showing a map as a representation given the generators.
As for the irreducibility of this representation, it should suffice to show there is no one-dimensional invariant subspace that satisfies the two matrices. Is that correct? We know the only invariant subspace of $p(12)$ is the span of (1,1) and (1,-1). However, since these are not invariant subspaces for $p(123)$, we have no invariant subspaces so this representation is irreducible.