# Joint Invariants of Linear Symplectic Actions
## 4 Variation on the Group and Space
### Symmetric Joint Invariants  
Invariants of the extended group \(\hat{G}=\operatorname{Sp}(2n,\mathbb{R})\times S_{m}\) on \(V^{\times m}\) are equivalent to \(G\)-invariants on configurations of unordered sets of points \(V^{\times m}/S_{m}\) (which is an orbifold). Denote the algebra of polynomial \(\hat{G}\)-invariants on \(V^{\times m}\) by \(\mathcal{S}_{G}^{m}\subset\mathcal{A}_{G}^{m}\). The projection \(\pi:\mathcal{A}_{G}^{m}\to\mathcal{S}_{G}^{m}\) is given by  
\[\pi(f)=\frac{1}{m!}\sum_{\sigma\in S_{m}}\sigma\cdot f.\]  
As a Noetherian algebra \(\mathcal{S}_{G}^{m}\) is finitely generated, yet it is not easy to establish its generating set explicitly. All linear terms average to zero, \(\pi(a_{ij})=0\), but there are several invariant quadratic terms in terms of the homogeneous decomposition \(\mathcal{A}_{G}^{m}=\oplus_{k=0}^{\infty}\mathcal{A}_{k}^{m}\). For example, for \(n=1\), \(m=4\) we have \(\mathcal{A}_{0}^{4}=\mathbb{R}\), \(\mathcal{A}_{1}^{4}=\mathbb{R}^{6}=\langle a_{12},a_{13},a_{14},a_{23},a_{24},a_{34}\rangle\,,\mathcal{A}_{2}^{4}=\mathbb{R}^{20}\) (21 monomials \(a_{ij}a_{kl}\) modulo 1 Plucker relation), etc. Then \(\pi(\mathcal{A}_{0}^{4})=\mathbb{R}\), \(\pi(\mathcal{A}_{1}^{4})=0\), and \(\pi(\mathcal{A}_{2}^{4})=\mathbb{R}^{2}\) has generators  
\[6\pi(a_{12}^{2}) = a_{12}^{2}+a_{13}^{2}+a_{14}^{2}+a_{23}^{2}+a_{24}^{2}+a_{34}^{ 2},\] \[12\pi(a_{12}a_{13}) = a_{12}a_{13}+a_{12}a_{14}+a_{13}a_{14}-a_{12}a_{23}-a_{12}a_{24} +a_{23}a_{24}\] \[+ a_{13}a_{23}-a_{13}a_{34}-a_{23}a_{34}+a_{14}a_{24}+a_{14}a_{34} +a_{24}a_{34}.\]  
**Theorem 2**. : _The field of symmetric rational invariants \(\mathfrak{F}_{G}^{m}=\pi(\mathcal{F}_{G}^{m})\) is the field of fractions \(\mathfrak{F}_{G}^{m}=F(\mathcal{S}_{G}^{m})\) and its transcendence degree is \(d(m,n)\)._  
Proof. : This follows from general theorems ([17], SS2.5) and discussion in Section 2. The last statement can be made more constructive: Let \(\ell\) numerate indices \((ij)\) of the basis \(\bar{\boldsymbol{a}}\) of \(\mathcal{F}_{G}^{m}\) as in Section 3.3, \(1\leq\ell\leq d=d(m,n)\). One can check that \(q_{k}=\pi(\prod_{\ell\leq k}a_{\ell}^{2})\) are algebraically independent. Thus, denoting \(\boldsymbol{q}=(q_{1},\dots,q_{d})\) we obtain the presentation  
\[\mathbb{R}(\boldsymbol{x},\boldsymbol{y})\supset\mathfrak{F}_{G}^{\text{alg} }\overset{d(m,n)}{\supset}\mathbb{R}.\]  
Here is an algorithm to obtain generators of \(\mathcal{S}_{G}^{m}\). **Proposition 1**. : _Fix an order on generators \(a_{ij}\) of \(\mathcal{A}_{G}^{m}\), and induce the total lexicographic order on monomials \(a^{\sigma}\in\mathcal{R}=\mathbb{R}[\boldsymbol{a}]\). Let \(\Sigma\) be the Grobner basis of the \(\mathcal{R}\)-ideal generated by \(\pi(a^{\sigma})\). Then elements \(\pi(a^{\sigma})\), contributing to \(\Sigma\), generate \(\mathcal{S}_{G}^{m}=\pi(\mathcal{A}_{G}^{m})\)._  
Proof. : Please note that the algorithm proceeds in total degree of \(a^{\sigma}\) until the Grobner basis stabilizes. That the involved \(\pi(a^{\sigma})\) generate \(\mathcal{S}_{G}^{m}\) as an algebra (initially they generate the ideal \(\mathcal{R}\cdot\pi(\mathcal{A}_{G}^{m})\subset\mathcal{A}_{G}^{m}\)) follows from the same argument as in the proof of [PERSON]'s theorem on invariants [1]. (The above \(\pi\) is the Reynolds operator used there.) Let us illustrate how this works in the first nontrivial case \(m=3\), for any \(n\).