$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}\DeclareMathOperator\SU{SU}$I'm a PhD student in physics working in the broad area of photonic quantum computing. My current project looks at the equivalence of any two $n$-photon $m$-mode Fock states under linear optical evolution.
These two Fock states can be rewritten as symmetric and homogenous polynomials of degree $n$ in $m$ variables. The linear optical evolution implies a linear $\SU(m)$ i.e. special unitary matrices of dimension $m$, change of basis for these variables. So basically, given any two Fock states, I'm interested in calculating the polynomial invariants of representations of the compact Lie group $\SU(m)$.
I've been going through the book ‘Algorithms in invariant theory’ by Bernd Sturmfels but it only mentions the first fundamental theorem for $\GL(m,\mathbb{C})$ and $\SL(m,\mathbb{C})$ Lie groups. Does there then exist a relation between the polynomial invariants of $\SU(m)$ and its complexification $\SL(m,\mathbb{C})$?
The only related result I could find was in the article ‘Lifting smooth homotopies of orbit spaces’ by Gerald Schwarz. The proposition (5.8) in it mentions the following algebra isomorphism:
\begin{equation} \mathbb{R}[W]^{\rho (K)} \otimes \mathbb{C} \simeq \mathbb{C}[V]^{\sigma (G)}, \end{equation}
where $K$ is a compact Lie group and $G = K_{\mathbb{C}}$ is its complexification. Their representations are also defined as $\rho: K \rightarrow \GL(W)$ and $\sigma: G \rightarrow \GL(V)$, where W is a real vector space and $V = W_{\mathbb{C}}=W\otimes_{\mathbb{R}}\mathbb{C}$.
I was hoping to get an easier description or examples of this algebra isomorphism if possible or any resources that point to the same. Also, is it possible to calculate the non-polynomial invariants that are related to the $L^{2}$ norm of the Fock state polynomials since they are also invariants under the $\SU(m)$ action?
