The integral shuffle algebra and the $K$-theory of the Hilbert scheme of points in $\mathbb{A}^2$.

We examine the shuffle algebra defined over the ring $\mathbf{R} = \mathbb{C}[q_1^{\pm 1}, q_2^{\pm 1}]$, also called the integral shuffle algebra, which was found by Schiffmann and Vasserot to act on the equivariant $K$-theory of the Hilbert scheme of points in the plane. We find that the modules of 2 and 3 variable elements of the integral shuffle algebra are finitely generated and prove a necessary condition for an element to be in the integral shuffle algebra for arbitrarily many variables.