The Group of Isometries of $K_0(\mathbb P_n)$

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 $n\leqslant 6$.