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.