Group of Isometries of the Lattice $K_0(\mathbb P_n)$

We study the group of isometries of the Grothendieck group $K_0(\mathbb P_n)$ which is equipped with a bilinear asymmetric Euler form. We prove several properties of this group; in particular, we show that it is isomorphic to the direct product of $\mathbb Z/2\mathbb Z$ and the free Abelian group of rank $[(n+1)/2]$. We also explicitly calculate its generators for $n\leqslant 6$.