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

We study the isometry group of the Grothendieck group \(K_0(\mathbb P_n)\) 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 \(\mathbb Z/2\mathbb Z\) by the free Abelian group of rank \([(n+1)/2]\) . We also explicitly calculate its generators for \(n\le 6\) .