8
ELDAR STRAUME
(B ) Diagrams A linear group (G.O) may be visualized by its diagram r(O), as
explained below. Write
( 2 ) (G,0) =
(U(1)r
x G'.o'i + + Ok), G' = G-| x x G
s
where Gj is simple and Oj is irreducible. r(O) has r + s vertices, namely the factors
U(1) and Gj. The following abridged notation will be used :
(n) = SO(n)
[n] = SU(n)
( 3 ) {n} = Sp(n)
@ = SX(n) = SU(n) or Sp(n)
= Sp(1) or SO(3)
o = U(1)
For each irreducible summand Oj of O, r(O) has a (q-l)-simplex whose vertices are
those q factors of G involved in the tensor product decomposition Oj = Y-j® ®^Q,
(^FJ nontrivial). We regard the simplex as a "q-fold link" between q vertices of type
U(1) or G;. In most cases the representations *Fj will be of the following types :
i) ( m ) p = (P
2
) P :U(1) - U(1) , z - * z P
») 5 n = Pn M-n. v n
Here 8
n
is the standard representation of SO(n), SU(n) or Sp(n) on [Rn, (Cn or Mn
respectively. The most typical simplices are 1-simplices, namely the 2-fold tensor
products :
( m ) (
n
)
:
Pm®KPn
[m] [n] : [ n
m
®0 M-nllR
( 4 ) {m} { n } : v
m
® n v
n
c ^
:
t ( ^ i )
p
® l . . ] R
o -
P
-
: (P2)P®K
••
The integer p e Z
+
in (4) is a twist coefficient ; such numbers appear when some
U(1) = SO(2) is involved in more than one summand Oj of O. Otherwise, we may
assume p = 1 (i.e. U(1) is effective) and then the label p is deleted from the diagram.
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