The following is mildly edited from the beginning of Bhargava's Higher Composition Laws I:
We will denote by $(\text{Sym}^k \mathbb{Z}^n)^*$ the lattice of $n$-ary $k$-ic forms with integer coefficients. The reason for the * is that there is also a sublattice $\text{Sym}^k \mathbb{Z}^n$ corresponding to the forms $f: \mathbb{Z}^n \rightarrow \mathbb{Z}$ satisfying $f(\xi) = F(\xi, \dots, \xi)$ for some symmetric multilinear function $F: \mathbb{Z}^n \times \cdots \mathbb{Z}^n \rightarrow \mathbb{Z}$ (classically called the `polarization' of $f$). Thus, for example, $(\text{Sym}^2 \mathbb{Z}^2)^*$ is the space of binary quadratic forms $f(x, y) = ax^2 + bxy + cy^2$ with $a, b, c \in \mathbb{Z}$, while $\text{Sym}^2 \mathbb{Z}^2$ is the subspace of such forms where $b$ is even.
This is enough to read and appreciate Bhargava's paper. That said, it seems that this should follow from standard definitions in representation theory, but I couldn't find these thoroughly treated in any of the reference books I have (e.g. Fulton-Harris).
It is possible to make some observations, for example, that $(\text{Sym}^k \mathbb{Z}^n)^*$ can naturally be constructed as a quotient of the tensor algebra, where $\text{Sym}^k \mathbb{Z}^n$ is a subspace. But it also seems to be possible to ask questions whose answers aren't obvious. For example, is the double dual of $\text{Sym}^k \mathbb{Z}^n$ isomorphic to $\text{Sym}^k \mathbb{Z}^n$? As abelian groups, sure, but working in the category of abelian groups would not preserve the distinction between $\text{Sym}^k \mathbb{Z}^n$ and its dual.
What are the intrinsic definitions, from which (I would guess) I could easily verify the above facts and answer my own question? Are these treated in some reference which I have overlooked?
Thank you.