# Numerical Treatment of Differential Equations by R. Bulirsch, R.D. Grigorieff, J. Schröder PDF

By R. Bulirsch, R.D. Grigorieff, J. Schröder

ISBN-10: 3540085394

ISBN-13: 9783540085393

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Let Zi (P ) be the ith term in the ascending central series; that is Z0 (P ) = 1 and for i > 1, Zi (P ) is the preimage in P of Z(P /Zi−1 (P )). Let Li (P ) be the ith term in the descending central series; that is L0 (P ) = P and for i > 0, Li (P ) = [Li−1 (P ), P ]. 1. Li (P ) ≤ Zc−i (P ). Proof. 6 in [2]. 2. [Li (P ), Zj (P )] ≤ Zj −i−1 (P ), where Zr (P ) = 1 for r ≤ 0. Proof. Let P ∗ = P /Zj −i−1 (P ). If i + 1 ≤ j then Zj (P )∗ = Zi+1 (P ∗ ) and Zj −i−1 (P )∗ = 1, while if i +1 ≥ j then P = P ∗ and Zj −i−1 (P ) = 1.

One obtains the following result. Theorem. Let p ≥ 3(h − 1) and λ ∈ X1 (T ). If either (a) G does not have underlying root system of type Cn (n ≥ 1) or (b) λ, αn∨ = p−2−c 2 , where αn is the unique long simple root and c is odd with 0 < |c| ≤ h − 1, then Ext 1Gσ (Fp ) (L(λ), L(λ)) = 0. Proof. 9 one has Ext1Gσ (Fp ) (L(λ), L(µ)) ∼ = ν∈ Ext 1G (L(λ) ⊗ L(ν)(r) , L(µ) ⊗ L(σ (ν))), h X(T )+ | ν, α0∨ where h = {ν ∈ < h}. The only complication in the quasi-split case versus the split case is that the weights ν and σ (ν) may be distinct.

R= ν∈ h By [BNP3, Thm. 4a], the remainder term R is exactly Ext 1G (L(λˆ ), L(µ)). 3. Various conjectures have been made about the dimensions of Ext1 -groups. Here it is shown that in most cases the dimensions of the Ext 1 groups between simple modules for the finite groups are bounded by the dimensions of Ext1 groups for the reductive algebraic groups. The corollary below is a generalization of [BNP3, Thm. 3]. The proof follows along the same lines and details are left to the reader. Corollary.