
236 Algebraic Operads: An Algorithmic Companion
total orders). For each n ≥ 1, let ω
n
∈ C be a primitive n-th root of unity,
and let v
n
= [1, ω
n
, . . . , ω
n−1
n
]. Write α =
1
2
(−1 +
√
−3) so that
v
1
= [1], v
2
= [1, −1], v
3
= [1, α, α], v
4
= [1, i, −1, −i].
Then clearly d(v
n
) = 1, 1, 2, 2 for n = 1, 2, 3, 4. For n = 5 the last 4 components
of v
5
are as follows where , η ∈ {±1} are independent signs:
1
4
−1 +
√
5 + η
√
−2
q
5 +
√
5
Hence d(ω
5
) = 4. See Exercise 7.15.
Lemma 7.4.1.9. For every v ∈ G
R
we have d(v) ≤ r = dim
Q
(G).
Proof. Let {v
1
, . . . , v
r
} be a basis for G over Q; this is also a basis for G
R
over R. Thus every v ∈ G
R
has the form v = a
1
v
1
+ ··· + a
r
v
r
for