Question about $\mathbb{Z}^2 \otimes \text{sym}^2\mathbb{Z}^3$.

Question

I am confused about the set $\mathbb{Z}^2 \otimes \text{sym}^2\mathbb{Z}^3$... Could someone please explain me why this corresponds to the set of pairs of symmetric $3$ by $3$ matrices? Thank you!

Answer 1

A standard basis for $S^2(\mathbb{Z}^3)$ is given by $(e_1e_1,e_1e_2,e_1e_3,e_2e_2,e_2e_3,e_3e_3)$, where juxtaposition denotes the symmetric product and $(e_i)_{i=1}^3$ is the standard basis of $\mathbb{Z}^3$. A symmetric integral $3 \times 3$ matrix is uniquely determined by $6$ integer entries which lie on the coordinates $(1,1),(1,2)(1,3),(2,2),(2,3),(3,3)$; therefore it is natural to identify the basis vector $e_ie_j$ with the basis matrix which has a single one in the $(i,j)$ position.

Finally, under this identification, the space of pairs of such matrices is $S^2(\mathbb{Z}^3) \oplus S^2(\mathbb{Z}^3) = \mathbb{Z}^2 \otimes S^2(\mathbb{Z}^3)$.