LOW COHOMOGENEITY ACTIONS 13
(b) r q : By Lemma 1.2, the linear group (G,0) extends to a linear group (G,I)
in the same c-equivalence class, where G = T9 x
G1.
(G,0) is a group of the type
discussed in subcase (a), or it is actually splitting. However, although r ( 0 ) may be
disconnected, we are looking for those subtori Tr eT°l which lead to connected diagrams
r(O), 0 = 0 |(T
r
xG'). An inclusion
Tr
e T°l is described by the "twist coefficients"
which appear in the diagram r(O). This procedure leads to the remaining linear groups
of Table III. To illustrate the above ideas we shall give two examples.
Example 1.3 : r = 2, q = k = 3. r ( 0 ) splits into three components, each of type
o (n) or c x ^ O
and one factor U(1) of T°l = U(1)3 is associated with each component. An inclusion
T
2
a T
3
is given by three
T2-weights
coj, say coj = a-frj + b#2 where {#j, i = 1,2 }
are unit weights with respect to a decomposition T
2
=
U(1)2.
To obtain r(E) from
r(O) each of the three simplices of r(O) defines a subdiagram of r(3), and they have
two vertices of type o in common. For example, a simplex o (n) of r(O) gives rise
to the diagram
n) or o (n) if b = 0
o
where the left picture indicates a 2-simplex, and the right one is a 1-simplex. r(O) is
the union of these three diagrams, and the coefficients a, b, . . should have values so that
the union is connected. The groups #45 through #47 in Table III are of this type.
Example 1.4 r ( 0 ) = o [n] o (#6a, Table III)
A circle group T
1
c T
2
is given by (atf-j -
b^2)-L,
a 0, (a,b) = 1. If b = 0 then
r(O) = o [ n l O #6b' T a b l e M| )
If a = b ( = 1) then (G,0) = (U(n),2p,n), but here c(O) = 4. For a * b and nonzero,
again c(O) = 3 and a, b are the "twist coefficients". Note the special case a = -b = 1
(#6c, Table III) :
1 _
T(O) = ^ 3 Z 3 ^
n ]
' ° = P2 ®IR [M-nllR = l m ®0 M-nllR
+
I *H ®0 M-nllR-
-1
) :
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