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)
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