Notations

We use the customary symbol s for sets of number s N C Z C Q C R C C, lettin g

N** C Zd Q • • - indicate tf-tuples a = (a

1

, ... , a

d

) o f suc h numbers.W e let (R d)+ denot e

the a € R

d

wit h af. 0 for all i, an d we let (Q

d)+

an d

(ZJ)+

b e the corresponding subset s

of Q

d

an d R

d.

W e notationally distinguis h th e dual vector space (R

d)*

fro m R

d,

lettin g

ev ... , e

d

an d e x, ... , e

d

b e the canonical base s for R

d

an d (R

d)*,

respectively.W e give

(Rd)*

th e dua l ordering

(Rd):={Z«/e,:«,o},

and we le t R d an d (R J)* have the dual norms

ItE^JL

= maxla/l

P

a/e/||j =

Ste/I-

Wecall A , = { / € (R*)*

:

/ 0 , \\f\\x = 1} th e fundamental d-simplex.

We say that a n r x s matri x \p = [fc

f

.A , K,.. £ R is positive i f fc

f

- 0 fo r all /, /, an d we

then write y 0 . Equivalently , if we regard y a s a linear map Rs — • Rr, we have a 0 im-

plies that if{a) 0 (a similar statement applie s to the dua l spaces).W e denote the transposed

matrix by p* and we identify i t with a matrix

(Rr)* —y

(R*) * in the usual way.

Finally we let GL(r , Z) denote the group of r x r matrices with entrie s in Z and de-

terminant ± 1, and we let G\Jj, Z)

+

b e the correspondin g matrice s with entries in Z

+.

I n

general y £ GL(r , Z)

+

doe s not impl y ip~

l

G GL(r, Z)

+.

http://dx.doi.org/10.1090/cbms/046/02