Let M be the $sl(n,C)$-representation of the inclusion $sl(n,C)\hookrightarrow gl(n,C)$. Let q be a symbol. $f(q)=1-M q + \wedge^2Mq^2-...+(-1)^n\wedge^nMq^n$

$g(q)=\sum_{i=0}^\infty Sym^iM \; q^i$

I want to prove that $f(q)g(q)=1$ which is equivalent to some isomorphism between many representations.

I am not sure if this choice of M is essential. Will any n-dim representation be enough? The worst method might be computing characters with character formulae...

btw product is tenor product. sum is direct sum.