  We derive an equation that is analogous to a well-known symmetric function
identity: $\sum_{i=0}^n(-1)^ie_ih_{n-i}=0$. Here the elementary symmetric
function $e_i$ is the Frobenius characteristic of the representation of
$\mathcal{S}_i$ on the top homology of the subset lattice $B_i$, whereas our
identity involves the representation of $\mathcal{S}_n\times \mathcal{S}_n$ on
the Segre product of $B_n$ with itself. We then obtain a q-analogue of a
polynomial identity given by Carlitz, Scoville and Vaughan through examining
the Segre product of the subspace lattice $B_n(q)$ with itself. We recognize
the connection between the Euler characteristic of the Segre product of
$B_n(q)$ with itself and the representation on the Segre product of $B_n$ with
itself by recovering our polynomial identity from specializing the identity on
the representation of $\mathcal{S}_i\times \mathcal{S}_i$.
