  We study the group of isometries of the Grothendieck group $K_0(\mathbb P_n)$
equipped with the standard Euler form $\chi$ defined by $\chi(E, F) =
\sum_{\nu}(-1)^\nu\dim Ext^\nu(E, F)$. We prove several properties of this
group, in particular, we show that it is essentially a free abelian group of
rank $[\frac{n+1}{2}]$. Also, we compute explicitly its generators for
$n\leqslant 6$.
