What is the dimension of this space? The function $π(v)$ interchanges the coordinates of the vector $v$ randomly.
For example:
$v = (1,3,7,9), π(v) = (7,3,1,9)$.
Fix some vector $v ∈ \mathbb{R}^n$ and construct a linear hull of all possible permutations of its coordinates.
What is the dimension of this space?
$\dim〈{π(v) | \text{ for all permutations } π}〉 = $ ?

How should I start to resolve this interesting challenge?
Are there some interesting hints for that?
Thanks.

Depending on $v$, the space can have dimension $0$, $1$, $n-1$ or $n$; but interestingly, once the dimension is known the vector subspace does not depend on the choice of $v$ if $n\geq 3$.
To see this, let us denote $V=\langle\{\pi(v) | \text{ for all permutations } \pi\}\rangle$. We have several cases to consider :


*

*$v$ has only zero coordinates. Then all $\pi(v)$ are zero as well, so $V=\{0\}$.

*all the coordinates of $v$ are equal and non-zero : then $\pi(v)=v$ for all permutations, and $V=\mathbb{R}\cdot (1,\dots,1)$.

*$v$ has two different coordinates : without loss of generality we can assume $v=(a,b,\dots)$ with $a\neq b$. Then $V$ also contains $(b,a, \dots)$, and also the difference $(a-b,b-a,0,\dots,0)$, and thus $(-1,1,0,\dots,0)$. Taking permutations on this vector, we get all vectors of the form $(-1,0,\dots,1,\dots,0)$, and taking linear combinations of these vectors, we can get any vector $x=(x_1,\dots,x_n)$ such that $\sum_j x_j=0$; indeed, it suffices to take the linear combination
\begin{gather} x_2(-1,1,0,\dots,0)+x_3(-1,0,1,0,\dots 0)+\dots +x_n(-1,0,\dots,0,1) \\ = (-x_2-x_3\dots -x_n, x_2, x_3,\dots, x_n)\\ = (x_1,x_2,\dots , x_n)\end{gather}(because $x_1=-x_2-\dots -x_n$).
Let us denote $W$ the vector subspace of such vectors. Since the sum of coordinates is invariant under the permutations, if $v\in W$ then $V\subset W$, and since we have shown that $W\subset V$, $V=W$. If $v\notin W$, then $V$ is bigger, and since $W$ has dimension $n-1$, $V=\mathbb{R}^n$.


Since every possible choice for $v$ falls under 1, 2 or 3, $V$ has to be either $\{0\}$, $\mathbb{R}\cdot (1,\dots,1)$, $W$, or $\mathbb{R}^n$, and thus $\dim V$ has to be either $0$, $1$, $n-1$, or $n$.

If you know what a linear representation of a group is, note that the problem is equivalent to determining the irreducible representations in the decomposition of the canonical representation of $S_n$, which is equivalent to determining the $\mathbb{R}[S_n]$-submodules of $\mathbb{R}^n$ seen as a $\mathbb{R}[S_n]$-module, which is precisely what I have done.
