spin representations and polynomials
I'm reading Group Theory and General Relativity by <PERSON> and his discussion of spin representations of SU(2) and the isomorphism to the space of homogenous polynomials is confusing me. I'll give some details, and point out my two questions (although an answer to Q1 might give me an answer to Q2).
A matrix $u\in SU(2)$ can be written as a transformation of a vector $\xi \in \mathbb{C}^2$: $$\bar{\xi} ^p=u^p_q\xi ^q$$ (summation convention assumed) Now we can consider representations of $SU(2)$ by forming products of these vectors in $\mathbb{C}^2\otimes\mathbb{C}^2\otimes...\otimes \mathbb{C}^2$. Such a vector would transform like $$\bar{\xi}^{p_1p_2...p_k}=u^{p_1}{q_1}u^{p_2}{q_2}...u^{p_k}_{q_k}\xi^{q_1q_2...q_k}$$
Where each $p_k$ or $q_k=1,2$. So this representation $u\rightarrow D^k(u)$ is not irreducible because there are symmetric vectors $\xi$ which are invariant under the action of all the $D^k(u)$. 656
Call this space of symmetric vectors $S_k$. Which bring me to...
Q1 Am I to understand these symmetric vectors have every entry proportional to the unit vector? So they look like
$$\xi=\alpha\binom{1}{1}\otimes...\otimes\binom{1}{1}$$
for some $\alpha$, so $S_k$ is one-dimensional?
This representation (which I think means $u\rightarrow D^k(u)$ together with $S_k$) forms the spin representation $Z_k$. The space $S_k$ can be found to be in 1-1 correspondence with the homogenous polynomials of degree $k$ in variables $z_1$ and $z_2$ with the following isomorphism
$$p(z_1,z_2)=\xi^{p_1p_2...p_k}z_{p_1}z_{p_2}...z_{p_k}$$
Q2: This is where I think my understanding of Q1 is incorrect, because it seems that these polynomials are of the form $$p(z_1,z_2)=\alpha (z_1^k+z_1^{k-1}z_2+...+z_2^k)$$ That's obviously homogenous, but it doesn't seem 1-1, because it doesn't seem that homogenous polynomials of the form $$\alpha_1 z_1^k+\alpha_2 z_1^{k-1}z_2+...+\alpha_{k+1}z_2^k$$ map back to anything in $S_k$.
Of course, the real interest here is in the form of $D^k(u)$ acting on these polynomials, but I can't get that far without figuring out what I don't understand here. Thanks in advance! 656