Suppose I have a finite-dimensional real vector space $X$ and a finite group $G$ that acts faithfully on X.
The task is to find a $G$-invariant basis of $X$. This means the set of basis vectors is invariant, not necessarily each basis vector for itself.
I suppose this pertains to representation theory, yet I have no knowledge about the subject and would like to learn about a place where to start. Can you determine whether such a basis exists, and how to compute it?
In fact, $G$ is just a symmetric group of permutations.
PS: In my application, $X$ is a subspace of the $p$-th symmetric product of $x_1,\dots,x_n$ with the $q$-th alternating product of $x_1,\dots,x_n$, and it is invariant under permutation of coordinates.
Let $G=\{1,-1\}$ act on $V$ by the usual multiplication by scalar. This action is obviously faithful. Let $v_1,\ldots,v_k$ be any basis. $-v_1$ depends linearly on $v_1$, but $-v_1\neq v_1$, hence $-v_1\not\in\{v_1,\ldots,v_k\}$, and $\{v_1,\ldots,v_k\}$ is not $G$-invariant.