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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?

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    It is easy to produce a messy formula for the character afforded by $K$. That gives then automatically the character of $K^{\otimes\ell}$. You can then calculate the inner product with the trivial charater, and be done with it. Something tells me you are looking for a formula in terms of $n$ and $\ell$ alone. This is a somewhat natural question, so it may well be that an answer is given in some text dedicated to representation theory of symmetric groups. Alas, I cannot point you at one. – 2017-01-31

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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 $p|n$.

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    (I write $V^{\oplus n}$ to distinguish it from the other possible interpretation $V^{\otimes n}$. However it is pretty clear you meant the former when writing $V^n$ since in the latter case $a$ wouldn't be a linear map.) – 2017-01-31
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    Thank you for your answer. In fact I would like to know the dimension of $( K^{\otimes l} )^{S_n}$, $(Sym^l K )^{S_n}$, and $(\wedge^l K )^{S_n}$ instead. Sorry for that, I edited my question. – 2017-01-31