Representation Decomposition of $ \operatorname{Sym}^{k+6} V$ In Fulton and Harris's Text Representation Theory: A First Course, exercise 1.12(b) asks to show that $\operatorname{Sym}^{k+6}V \cong \operatorname{Sym}^kV \oplus R$ as representations of $\frak S_3$. $V$ is the 2-dimensional standard representation of $\frak S_3$ and $R$ is the regular representation. Note that this is supposed to be done without using character theory.
The hint given by my professor was to show that $\operatorname{Sym}^6 V \cong U \oplus R \cong U^{\oplus 2} \oplus U' \oplus V^{\oplus 2}$, where $U$ is the trivial representation and $U'$ is the alternating representation. Then, we're supposed to use that to find copies of $\operatorname{Sym}^k V$ and $R$ in $\operatorname{Sym}^{k+6} V$ that intersect only at $0$, which will then prove the original isomorphism.
Proving the congruence was relatively straightforward, but I have no idea how we're supposed to find copies of $\operatorname{Sym}^k V$ and $R$ in $\operatorname{Sym}^{k+6} V$. 
 A: This exercise makes a lot more sense in the context of the book. In fact it's probably one of the few problems about representations of finite groups for which the approach in that chapter is more suitable than character theory :-). (The authors themselves state that the approach is "superfluous" for finite groups and they're only using it to introduce an idea that's useful for Lie groups.)
So, using the notation and the basis $\alpha,\beta$ of $V$ from that chapter, the elements of $\text{Sym}^kV$ can be paired up as $\alpha^j\beta^{k-j}$ and $\alpha^{k-j}\beta^j$ (except for $\alpha^{k/2}\beta^{k/2}$ for even $k$). The transposition $\sigma$ transforms these into each other, and their eigenvalues under $\tau$ are $\omega^j\omega^{2(k-j)}=\omega^{2k-j}=\omega^{-(k+j)}$ and $\omega^{k-j}\omega^{2j}=\omega^{k+j}$, respectively. Thus, if $k+j\equiv0\pmod3$, they form a trivial and an alternating representation, and if $k+j\not\equiv0\pmod3$ they form the standard representation. Thus, if $k\gt6$, there are three successive values of $j$ for which the three pairs together form two standard, one alternating and one trivial representation, i.e. the regular representation. The remaining values of $j$ have the same remainders mod $3$ as the values for $k-6$, which implies $\text{Sym}^{k+6}V \cong \text{Sym}^kV \oplus R$.
