Dimension of an invariant subspace Let $V$ be a vector space of dimension $k$ and $n$ a positive integer. Let $a: V^n \to V$ be the map sending $(v_1,\ldots,v_n)$ to $v_1 + \cdots + v_n$. The symmetric group $S_n$ acts on $V^n$ by permuting the factors and the subspace $K := ker(a)$ is stable under this action.
What is the dimension of the vector spaces $( K^{\otimes l} )^{S_n}$, $(\wedge^lK)^{S_n}$, and $(Sym^lK)^{S_n}$ in terms of $k$, $n$, and $l$?  Is there any reference which does systematically this kind of computations?

UPDATE: this answer responded to an earlier version of the question, see comments below.
Any element of $V^{\oplus n}$ (this includes elements of $K$) that is invariant under $S_n$ is of the form $(v, v, \ldots, v)$. Now for such an element to lie in $K$ we must have $v + v + \ldots + v = 0$ ($n$ terms). If the characteristic of the ground field over which you have your vectorspace $V$ is zero (i.e. if the ground field are the complex numbers) this can only happen when $v = 0$ and so $K^{S_n} = \{0\}$.
If the characteristic of the ground field is $p$ then $v + v + \ldots + v = 0$ is possible when $p|n$. However, in that case it is not only possible but also inevitable. It follows  that for fields of characteristic $p$ the answer is:
$K^{S_n} = \{0\}$ if $p \not|  n$ and
$K^{S_n} = (V^{\oplus n})^{S_n} = \{(v, \ldots, v) : v \in V \}$ if $p|n$. 
