Tensor product with a 1-dimensional representation

Let be an arbitrary 1-dimensional representation and be an irrep over . Then is an irrep. #m/thm/rep

Proof

For any where is a subspace, , Hence the invariant subspaces of are the same as those of , and thus if is irreducible so is .

As a result, 1-dimensional irreps form a group.

#state/tidy | #lang/en | #SemBr