In this chapter, we investigate the following question. Suppose we have a geometrically irreducible middle extension sheaf G on Gm/k which is pure of weight zero, such that the object N := G(1/2) ∈ Parith has “dimension” n and has Ggeom,N = Garith,N = GL(n). Suppose in addition we are given s ≥ 2 distinct characters χi of k×. We want criteria which insure that for the objects
Ni := N ⊗ Lχi,
the direct sum ⊕iNi has Ggeom,⊕iNi = Garith,⊕iNi = ∏i GL(n). Because we have a priori inclusions Ggeom,⊕iNi ⊂ Garith,⊕iNi ⊂ ∏i GL(n), it suffices to prove that Ggeom,⊕iNi = ∏i GL(n). To show this, it suffices to show both of the following two statements.
(1) The determinants in the Tannakian sense “det”(