698 7. ELIMINATING CASES CORRESPONDING TO NO SHADOWE.3.37; by 7.3.4 this column will give an upper bound on w. Column 6, labeled"m 2 :S:", gives an upper bound on m 2 := m2(Mv ). Columns 7 and 8, labeled
"/3 ?:" and a ?:", give the minimum codimension of a subspace U of V suchthat 02 (CM(U)) 1:. CM(V), or such that Cfifv(U) contains an (F - 1) offender,
respectively. If there are no ( F -1 )-offenders, then Ji (T) centralizes V and column
8 contains oo. We remark that the minimum of a and /3 by 7.4.1 gives a lower
bound for the parameter r of Definition E.3.3 in the cases where L ::::! M.
PROPOSITION 7.2.1. The values of various parameters for our modules are:
case I a:::; m?: w ?: n' m2:::; /3?: a?:
SU3(2n)/6n n 2n n n n+l 4n 00Sz(2n)/4n n 2n n n n 3 fin 00
(S)L3(2^2 n)/9n 3n 3n 0 2n 4n 4n oo; 5ifn=1
M12/l0^2 4 2 2 4 6 00
3M2z/12 3 4 1 2 5 8 00
M22/l0^3 3 0 2 5 6 6
M22/l0 3 3 0 2 5 6 5M23/ll 3 4 1 2 4 6 00
M23/ll 3 4 1 2 4 6 5
M24/ll 3 4 1 2 6 6 7
M24/ll^3 4 1 2 6 6 5
SL3(2n).2/3n EEl 3n n 2n n n 2n 4n oo;2 if n = 1
Sp4(2n)'.2/4n EEl 4nf < 2n 3n >n n 3n 4n 0087/4 EEl 4 2 4 2 2 3 4 00
L3(2) 12/3 Q9 3t (^2 3 1 2 4 6 3)
Lz(2n) 12/2n Q9 2nf (n) n 0 n 2n (2n) oo; 2 if n = 1
7.3. Bounds on w
We now implement the outline discussed in subsection E.3.3.As remarked earlier, in chapter 7 and the next chapter 8, we exclude the final
case in the Tables of Propositions 7.1.1 and 7.2.1:HYPOTHESIS 7.3.1. V is not the orthogonal module for Lo~ nt(2n).
Recall that the case excluded by Hypothesis 7.3.1 will be treated by other
methods in the third chapter 9 of this part 3. Thus in this chapter and the next,
discussion of "all" cases in the Tables refers to the remaining cases, with the final
row of the Tables excluded.We first discuss the parameters r and s. See Definitions E.3.3, E.3.5, E.3.1,
and E.3.9 for the parameters r, s, m, and a.PROPOSITION 7.3.2. r ?: m, so that s = m.PROOF. This follows from Theorem E.6.3 when m > 2, which we see from
Table 7.2.l holds in all cases except for L3(2) on 3 EEl 3. In that case we make
a direct argument, but as the methods are of a different flavor from the uniform
treatment in this chapter, we banish those details to a mini-Appendix at the end
of the chapter; see 7.7.1 for the proof. D