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 $\chi (E,F)={\sum }_{\nu }(-1{)}^{\nu }\dim Ex{t}^{\nu }(E,F)$. 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 $n⩽6$.