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By Berman P.H.

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9, compute ri = dimC Ti , i where Ti = (ci,1 , . . , ci,νi ) : the system Z = Mi Z + ci,j Bi,j j admits a k-rational solution . Let r˜i = νi − ri . 3. Using [CS99], compute a set H of defining equations for Ψ(GH ), where Ψ : GH → GLn (C) is a matrix representation of GH on VH with respect to some basis of VH having the following property: Given a matrix Q = Ψ(σ), σ ∈ GH , then we have ¯i, . . , Q ¯ i ) (νi copies) and Q ¯ i gives the action Q = diag(Q1 , . . , Qs ) with Qi = diag(Q of σ on VMi with respect to some fixed basis of VMi for 1 ≤ i ≤ s.

J rl,mj vl,j , . . , j rs,νs j vs,j ). j The first statement of the conclusion of the lemma follows immediately. The second statement then follows after defining S ⊆ C ν = C ν1 to be the row space of the matrix R = R1 . ˜ +B ˜ such that the We are interested in inhomogeneous systems of the form Y = AY ˜ is completely reducible. A system having this associated homogeneous system Y = AY property is equivalent to a system of the form = AY + B, A = diag(A1 , A2 , . . , As ), B = (B1T , B2T , .

Therefore there exist S˜ and L˜2 ∈ D such that ˜+L ˜ 2 L2 = 1. ˜ = SL ˜ 2 and RR L1 R ˜ 1 = 1. We have that ˜ + L1 L We now claim that SS ˜ 1 )L1 ˜ + L1 L (SS = ˜ 1 + L1 L ˜ 1 L1 SSL = ˜ 2 R + L1 (1 − RR) ˜ SL = ˜ 2 R + L1 − L1 RR ˜ SL = ˜ 2 R + L1 − SL ˜ 2R SL = L1 , and the equation follows after cancelling L1 on the right. To prove one direction of the lemma, suppose L1 (f ) = b for some f ∈ k. If h = R(f ) ∈ k, then L2 (h) = SL1 (f ) = S(b) as desired. To prove the other direction, suppose L2 (h) = S(b) ˜ ˜ 1 (b) ∈ k.

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