We study the action of the Klein simple group PSL2(F7 ) consisting of 168 elements on two rational threefolds: the three-dimensional projective space and a smooth Fano threefold X of anticanonical degree 22 and index 1. We show that the Cremona group of rank three has at least three non-conjugate subgroups isomorphic to PSL2(F7 ). As a by-product, we prove that X admits a Kähler–Einstein metric, and we construct a smooth polarized K3 surface of degree 22 with an action of the group PSL2( F7 ).
Unless explicitly stated otherwise, varieties are assumed to be projective, normal and complex.
We use the theory of tilting modules for algebraic groups to propose a characteristic free approach to "Howe duality" in the exterior algebra. To any series of classical groups (general linear, symplectic, orthogonal, or spinor) over an algebraically closed field k, we set in correspondence another series of classical groups (usually the same one). Denote by G1 (m) the group of rank m from the first series and by G2 (n) the group of rank n from the second series. For any pair (Ga(m),G2(n)) we construct the Gl(m) x G2(n)-module M(m,n). The construction of M(m, n) is independent of characteristic; for char k -~ 0, the actions of Gl(m) and G2(n) on M(m,n) form a reductive dual pair in the sense of Howe.