  Let $K$ be a field with ${\rm char}(K)=0$. For a partition $\lambda$ of $n
\in {\mathbb N}$, let $I^{\rm Sp}_\lambda$ be the ideal of
$R=K[x_1,\ldots,x_n]$ generated by all Specht polynomials of shape $\lambda$.
These ideals have been studied from several points of view (and under several
names). Using advanced tools of the representation theory, Berkesch Zamaere et
al [BGS]. constructed a minimal free resolution of $I^{\rm Sp}_{(n-d,d)}$
except differential maps. The present paper constructs the differential maps,
and also gives an elementary proof of the result of [BGS].
