# Joint Invariants of Linear Symplectic Actions
## 1 Introduction  
The classical invariant theory [1; 2; 3] investigates polynomial invariants of linear actions of a Lie group \(G\) on a vector space \(V\), i.e., describes the algebra \((SV^{*})^{G}\). For instance, the case of binary forms corresponds to \(G=\mathrm{SL}(2,\mathbb{C})\) and \(V=\mathbb{C}^{2}\); equivalently for \(G=\mathrm{GL}(2,\mathbb{C})\) one studies instead the algebra of relative invariants. The covariants correspond to invariants in the tensor product \(V\otimes W\) for another representation \(W\). Changing to the Cartesian product \(V\times W\) leads to joint invariants of \(G\).  
In this paper, we discuss joint invariants corresponding to the (diagonal) action of \(G\) on the iterated Cartesian product \(V^{\times m}\) for increasing number of copies \(m\in\mathbb{N}\). We will focus on the case \(G=\mathrm{Sp}(2n,\mathbb{R})\), \(V=\mathbb{R}^{2n}\) and discuss the conformal \(G=\mathrm{CSp}(2n,\mathbb{R})=\mathrm{Sp}(2n,\mathbb{R})\times\mathbb{R}_{+}\) and affine \(G=\mathrm{ASp}(2n,\mathbb{R})=\mathrm{Sp}(2n,\mathbb{R})\times\mathbb{R}^{2n}\) versions later.  
This corresponds to invariants of \(m\)-tuples of points in \(V\), i.e., finite ordered subsets. By the [PERSON] [1] and [PERSON] [4] theorems, the algebra of polynomial invariants (for the semi-simple \(G\)) or the field of rational invariants (in all other cases considered) can be interpreted as the space of functions on the quotient space \(V^{\times m}/G\).  
For \(G=\mathrm{Sp}(2n,\mathbb{C})\) the algebra of invariants is known [5]. Generators and relations (syzygies) are described in the first and the second fundamental theorems, respectively. We review this in Theorem 1 (real version), and complement by explicit examples of free resolutions of the algebra. In addition, we describe the field of rational invariants.  
We also discuss invariants with respect to the group \(G=\mathrm{Sp}(2n,\mathbb{R})\times S_{m}\), in which case considerably less is known. Another generalization we consider is the field of invariants for the conformal symplectic Lie group \(G=\mathrm{CSp}(2n,\mathbb{R})\) on the contact space.  
When approaching invariants of infinite sets, like curves or domains with smooth boundary, the theory of joint invariants is not directly applicable and the equivalence problem is solved via differential invariants [6]. In the case of a group \(G\) and a space \(V\) as above this problem was solved in [7]. We claim that the differential invariants from this reference can be obtained in a proper limit of joint invariants, i.e., via a certain discretization and quasiclassical limit, and demonstrate it explicitly in several cases.  
In this paper, we focus on discussion of various interrelations of joint invariants. In particular, at the conclusion we note that joint invariants can be applied to the equivalence problem of binaryforms. Since these have been studied also via differential invariants [2; 8] a further link to the above symplectic discretization is possible.  
The relation to binary forms mentioned above is based on the Sylvester theorem [9], which in turn can be extended to more general Waring decompositions, important in algebraic geometry [10]. Our computations should carry over to the general case. This note is partially based on the results of [11], generalized and elaborated in several respects.