  We study Le Potier's strange duality conjecture on $\mathbb{P}^2$. We focus
on the strange duality map $SD_{c_n^r,d}$ which involves the moduli space of
rank $r$ sheaves with trivial first Chern class and second Chern class $n$, and
the moduli space of 1-dimensional sheaves with determinant
$\mathcal{O}_{\mathbb{P}^2}(d)$ and Euler characteristic 0. By using tools in
quiver representation theory, we show that $SD_{c^r_n,d}$ is an isomorphisms
for $r=n$ or $r=n-1$ or $d\leq 3$, and in general $SD_{c^r_n,d}$ is injective
for any $n\geq r>0$ and $d>0$.
