I am at best superficially acquainted with the intricacies of Hodge theory. The following question comes from my study of the paper On the $\Gamma$-factors attached to motives by Christopher Deninger. In chapter §3 the auther defines $\mathbb{R}(1)$ to be the real Tate-Hodge structure where $\mathbb{R}(1)=\mathbb{Z}(1)\otimes_{\mathbb{Z}}\mathbb{R}$ and $\mathbb{Z}(1)$ is the unique pure one-dimensional Tate structure of weight $-2$.
He then speaks of the "infinite real Hodge structures" \begin{equation*}\begin{matrix}\operatorname{Sym}\mathbb{R}(1) & \text{and} & \operatorname{Sym}\mathbb{R}(1)[\mathbb{R}(-1)]\end{matrix}\text{.}\end{equation*}
I assume $\operatorname{Sym}$ denotes the symplectic group? How are these structures defined?