ANTHONY V. PHILLIPS and DAVID A. STONE

4

s

*

i

T ^

!M./^0

• A

Figure 1.1: The standard projection 717: C r — * Ar

Definition 1.3 We define the standard projection

7rf:Cr

—

Ar

by

induction on r; see Figure 1.1. For r = 0, the definition is trivial. Now

assume that 7r^7__1: Cr"1 — e

0

,..., er_i has been defined. We iden-

tify Cr~1 with drCr and set 7r^(,Si,... ,«sr) = (1—5r)7r^_1(si,... ,sr_i) +

In terms of coordinates, ft*?(s\,..., sr) = (t0, • • •, ^r), where

tQ = ( l _

5 l

) ( l _

5 2

) . . . ( l _

5 r

)

! = 5 i ( l - 5

2

) " - ( l - S

r

)

2 = S2'-(l-Sr)

tr =

1.4 The standard subdivision of C

r

. For every permutation a of

{ 1 , . . . , r} we define an r-simplex in Cr by the rule (see Figure 1.2)