Let $V_d$ be the vector space of homogeneous polynomials of degree $d$ in three variables $x, y,$ and $z$, and let the symmetric group $S_3$ act on $V_d$ by permuting the variables. Find the irreducible subrepresentations of $V_d$ in case $d = 1, 2,$ and $3.$
I am confused about what this question is actually asking. In the case where $d=1$ we have the space of polynomials of the form $ax+by+cz$. The group $S_3$ will act by permuting the variables. Considering the polynomials as vectors in the $x,y,z$ basis we can write them as $(a,b,c)$ and then the six permutations of $S_3$ map to linear transformations permuting the basis. So what does it mean to find the irreducible subrepresentations? I though an irreducible subrepresentation was something you have on the group, i.e. one of the three irreducible subrepresentations of $S_3$ is the trivial representation on a one dimensional vector space where all of $S_3$ is sent to the identity. Why am I being asked to find the irreducible representations the vector space (rather than the group)?