194
8 Structures Relevant to Physics
are order-preserving. The map τ interchanges the two middle tensor factors. In
particular, if i = n 1 + 1, (8.16) is the summation over all (n 1 , n 2 )-unshuffles ρ ∈
Σ n 1 +n 2 , i.e. permutations ρ ∈ Σ n 1 +n 2 such that
ρ(1) < · · · < ρ(n 1 ) and ρ(n 1 + 1) < · · · < ρ(n 1 + n 2 ).
Proof. Consider the map θ : [n+2] → [n]]{i, j } which coincides with τ ij of (6.69)
on [n + 2] \ {i, j }, while θ(i) := i and θ(j ) := j . By the functoriality (6.53) one
has the equality
◦ ij M (θ ) = M (τ ij )◦ ij
of maps M (n + 2; g) → M (n; g). On the other hand, M (τ ij )◦ ij = ◦ ij by the
definition (6.70) of skeletal operations, thus ◦ ij M (θ ) = ◦ ij . Likewise we establish
that • ij T (θ ) = • ij . Formula (8.15) is then obtained by taking in (8.2) the above
isomorphism θ .
Let us prove (8.16). Recall that the summation (8.3) defining the bracket runs
over all subsets S 1 , S 2 ⊂ [n 1 + n 2 ] such that S 1 S 2 = [n 1 + n 2 ] and |S 1 | = n 1 ,
|S 2 | = n 2 . Let X 1 , X 2 be two such subsets. There clearly exists a one-to-one
correspondence between couples (S 1 , S 2 ) and automorphisms ρ : [n 1 +n 2 ]
∼ =
− → [n 1 +
n 2 ] whose restrictions to X 1 and X 2 are order-preserving. It follows from this
observation and the functoriality of the structure operations a ◦ b and a • b that, for
fixed bijections
ϑ 1 : [n 1 + 1]
∼ =
− → X 1 {a} and ϑ 2 : [n 2 + 1]
∼ =
− → X 2 {b}
(8.17)
the right-hand side of (8.3) equals the sum
ρ
M (ρ) ⊗ T (ρ)
( a ◦ b ⊗ a • b )
ϑ 1 ⊗ ϑ 2 ⊗ ϑ 1 ⊗ ϑ 2 )τ (g ⊗ h)
(8.18)
over automorphisms ρ : [n 1 + n 2 ]
∼ =
− → [n 1 + n 2 ] as above.
Let κ ij be as in (6.7) with m := n 1 and n := n 2 . Take
X 1 := {1, . . . , i − 1} ∪ {n 2 + i, . . . , n 1 + n 2 }, X 2 := {i, . . . , i + n 2 − 1}
and notice that
X 1 = κ ij
[n 1 + 1] \ {i}
and X 2 = κ ij
[m 1 + 1] \ {j }
.
Take as ϑ 1 in (8.17) the isomorphism that restricts to κ ij on [n 1 + 1] \ {i} and
sends i ∈ [n 1 + 1] to a. The isomorphism ϑ 2 is defined similarly. Recalling the
definition (6.8) of the skeletal operations, we see that
a ◦ b (ϑ 1 ⊗ ϑ 2 ) = M (κ ij ) i ◦ j = i ◦ j
8 Structures Relevant to Physics
are order-preserving. The map τ interchanges the two middle tensor factors. In
particular, if i = n 1 + 1, (8.16) is the summation over all (n 1 , n 2 )-unshuffles ρ ∈
Σ n 1 +n 2 , i.e. permutations ρ ∈ Σ n 1 +n 2 such that
ρ(1) < · · · < ρ(n 1 ) and ρ(n 1 + 1) < · · · < ρ(n 1 + n 2 ).
Proof. Consider the map θ : [n+2] → [n]]{i, j } which coincides with τ ij of (6.69)
on [n + 2] \ {i, j }, while θ(i) := i and θ(j ) := j . By the functoriality (6.53) one
has the equality
◦ ij M (θ ) = M (τ ij )◦ ij
of maps M (n + 2; g) → M (n; g). On the other hand, M (τ ij )◦ ij = ◦ ij by the
definition (6.70) of skeletal operations, thus ◦ ij M (θ ) = ◦ ij . Likewise we establish
that • ij T (θ ) = • ij . Formula (8.15) is then obtained by taking in (8.2) the above
isomorphism θ .
Let us prove (8.16). Recall that the summation (8.3) defining the bracket runs
over all subsets S 1 , S 2 ⊂ [n 1 + n 2 ] such that S 1 S 2 = [n 1 + n 2 ] and |S 1 | = n 1 ,
|S 2 | = n 2 . Let X 1 , X 2 be two such subsets. There clearly exists a one-to-one
correspondence between couples (S 1 , S 2 ) and automorphisms ρ : [n 1 +n 2 ]
∼ =
− → [n 1 +
n 2 ] whose restrictions to X 1 and X 2 are order-preserving. It follows from this
observation and the functoriality of the structure operations a ◦ b and a • b that, for
fixed bijections
ϑ 1 : [n 1 + 1]
∼ =
− → X 1 {a} and ϑ 2 : [n 2 + 1]
∼ =
− → X 2 {b}
(8.17)
the right-hand side of (8.3) equals the sum
ρ
M (ρ) ⊗ T (ρ)
( a ◦ b ⊗ a • b )
ϑ 1 ⊗ ϑ 2 ⊗ ϑ 1 ⊗ ϑ 2 )τ (g ⊗ h)
(8.18)
over automorphisms ρ : [n 1 + n 2 ]
∼ =
− → [n 1 + n 2 ] as above.
Let κ ij be as in (6.7) with m := n 1 and n := n 2 . Take
X 1 := {1, . . . , i − 1} ∪ {n 2 + i, . . . , n 1 + n 2 }, X 2 := {i, . . . , i + n 2 − 1}
and notice that
X 1 = κ ij
[n 1 + 1] \ {i}
and X 2 = κ ij
[m 1 + 1] \ {j }
.
Take as ϑ 1 in (8.17) the isomorphism that restricts to κ ij on [n 1 + 1] \ {i} and
sends i ∈ [n 1 + 1] to a. The isomorphism ϑ 2 is defined similarly. Recalling the
definition (6.8) of the skeletal operations, we see that
a ◦ b (ϑ 1 ⊗ ϑ 2 ) = M (κ ij ) i ◦ j = i ◦ j