Symmetric and exterior powers over an arbitrary field

Let $V$ be a vector space over an arbitrary field $k$ and let $n\ge 0$. Permuting tensor factors gives an action of the symmetric group $S_n$ on $V^{\otimes n}$. Define

$S^n(V)=V^{\otimes n}\mathrm{/}{R}_n,$

where $R_n$ is the linear span of all $t-\pi t$ with $t\in V^{\otimes n}$ and $\pi \in S_n$. Thus $S^n(V)$ is the quotient by coinvariance relations, not the subspace of invariant tensors.

The exterior power ${\mathrm{\Lambda }}^n(V)$ is the repeated-vector quotient of The $k$th exterior power as the tensor-power quotient by repeated-vector relations: it is $V^{\otimes n}$ modulo the span of pure tensors in which two inputs are equal.

By convention $V^{\otimes 0}=k$, so $S^0(V)={\mathrm{\Lambda }}^0(V)=k$, while $S^1(V)={\mathrm{\Lambda }}^1(V)=V$. These definitions require no division by $n!$ and therefore apply in every characteristic.