Let (, V) be a representation of a group G, where V is a vector space of dimension n. Suppose that its character, , is the constant function n:

(g) = n, g G.

Prove that is a trivial representation. Namely, that (g) = IdV for all g G

I know this question has previously been asked, but that answer is erroneous (character is not multiplicative when the dimension is bigger than 1) which is why I am asking again. Thank you for answering in advance.