7.2. PARAMETERS FOR THE REPRESENTATIONS 697Mv ~ I restr. on n dimV descr. V shadows example
U3(2n) n~2 6n natural
Sz(2n) odd n ~ 3 4n natural
L3(22n) n~2 9n 3 Q9 30- U5(2n), U7(2n)
Aut(M12) 10 irred. perm.
M22^12 unitary
M22^10 code Co2
10 cocode F22
M23^11 code
11 cocode F23
M24^11 code Co1
11 co code F24 J4
SL3(2n).2 6n 3n E9 3nSp4(2n)'.2 8n 4n E9 4nt
87 8 4E94
£3(2) 12 9 3 Q9 3t £5(2).2, £7(2).2L2(2n) 12 n~2 4n 2n Q9 2nt L4(2n ).2, L5 (2n ).2
7.2. Parameters for the representations
Our main task in chapter 7 will be to eliminate the cases not corresponding
to a shadow or example. We use the weak closure methods of section E.3. These
methods are "numerical", in the sense that they compare parameters-such as
a, m, n', a, f3 determined only by the representation of Mon V, and on other pa-
rameters r, s, w determined by suitable subspaces U of V with Ca(U) ::::; M. We will
obtain a numerical contradiction from the Fundamental Weak Closure Inequality
involving these parameters, established in E.3.29.^1
Because the initial steps in the weak closure argument involve primarily the
parameters m 2 of Mv and m, a of the module V, estimates on these values are
included in the early columns of the Table in Proposition 7.2.1 below.
Proofs that the parameters are indeed as indicated in the Table appear in cor-
responding .sections of chapter H of Volume I-with the exception of the parameter
n', which is determined in 7.3.4. Certain values in the table are given in parenthe-
ses; these are values which seem to be well known, but which we do not require in
our argument, and hence are not verified in chapter H. The last two columns of the
table list parameters a and f3 primarily relevant to an application of E.6.27 later in
this chapter; the derivation of these parameters also appears in chapter H, except
in some cases like the last case where they are not used.
We now describe the Table in more detail: Column 1, labeled "case", indicates
the pair L 0 , V discussed in the corresponding row. Column 2, labeled "a::::;", gives
an upper bound on a.:= a(Mv, V). Column 3, labeled "m ~",gives a lower bound
on m := m(Mv, V). The definitions of these parameters appear as E.3.9 and E.3.1.
Column 4, labeled "w ~", gives the resulting lower bound on the difference m - a,
which is in turn a lower bound on the parameter w of Definition E.3.23 by 7.3.3.
Column 5, labeled "n'", is the parameter n' := n'(Auta(V)) given in Definition
(^1) 0f course, local configurations L, V that actually exist in shadows are not eliminated nu-
merically. So in the following chapter 8, we instead show that those configurations provide the
unique solution to the FWCI; and then eliminate the cases by showing those configurations violate
our SQTK hypothesis.