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6 Operads
and = (τ ) the Koszul sign of the permutation
τ : v 1 , . . . , v n −→ v 1 , . . . , v i−1 , v i+n , . . . , v m+n ,
v n+i−j +1 , . . . , v i+n−1 , v i , . . . , v n+i−j .
Let us explain how the sign in (6.25) appears. To save the space, we denote, only
for the purpose of this explanation,
V 1 := V
⊗(i−1) , V 2 := V
⊗(n−j +1) , V 3 := V
⊗(j −1) , and V 4 := V
⊗(m−i+1) .
We also denote
ω 1 := v 1 ⊗ · · · ⊗ v i−1 ∈ V 1 , ω 2 := v i ⊗ · · · ⊗ v n+i−j ∈ V 2 ,
ω 3 := v n+i−j +1 ⊗ · · · ⊗ v n+i−1 ∈ V 3 and ω 4 := v i+n ⊗ · · · ⊗ v m+n ∈ V 4 .
With this notation, (6.26) reads as
κ = |f ||g| + |s
i ||ω 4 | + |s
i ||ω 3 |.
It follows from the definition of the a ◦ b -operations in the endomorphism operad
given in Example 6.6 that the skeletal f i ◦ j g is the composition of the permutation
τ : V 1 ⊗ V 2 ⊗ V 3 ⊗ V 4 −→ V 1 ⊗ V 4 ⊗ V 3 ⊗ V 2
followed by the isomorphism
V 1 ⊗ V 4 ⊗ V 3 ⊗ V 2
∼ =
−→ V 1 ⊗ V 4 ⊗ k ⊗ V 3 ⊗ V 2
(6.27)
and then by
(1
⊗m
⊗ s ⊗ 1
⊗n ) : V 1 ⊗ V 4 ⊗ k ⊗ V 3 ⊗ V 2 −→ V 1 ⊗ V 4 ⊗ V ⊗ V ⊗ V 3 ⊗ V 2
followed by the permutation
ρ : V 1 ⊗ V 4 ⊗ V ⊗ V ⊗ V 3 ⊗ V 2 −→ V 1 ⊗ V ⊗ V 4 ⊗ V 3 ⊗ V ⊗ V 2
and, finally, composed with
f ⊗ g : V 1 ⊗ V ⊗ V 4 ⊗ V 3 ⊗ V ⊗ V 2 −→ k.
Let us inspect how the composition of the above maps acts on the element
v 1 ⊗ · · · ⊗ v n+m = ω 1 ⊗ ω 2 ⊗ ω 3 ⊗ ω 4 ∈ V
⊗(m+n) .