Given a vector space $V$ of dimension $n$, I can form the module $$ S^{(1^{nd})}(V)$$ where $(1^{nd})$ denotes the partition $$ (1,1,\dots,1)\vdash nd$$ Now for $d=1$, I know that this module is in fact $\bigwedge^n(V)$. However I don't understand what it is for $d>1$. Is it still irreducible as a $GL(V)$-module? Does it equal $(\bigwedge^n(V))^{\otimes d}$?
Thanks in advance.