Annales de l’institut Fourier (AIF)

Matheus, Carlos

Yoccoz, Jean-Christophe

Zmiaikou, David

AN
N
A
L
E
S
D
E
L’INSTI
T
U
T
F
O
U
R
IE
R
ANNALES
DE
L’INSTITUT FOURIER
Carlos MATHEUS, Jean-Christophe YOCCOZ & David ZMIAIKOU
Corrigendum to “Homology of origamis with symmetries”
Tome 66, no 3 (2016), p. 1279-1284.
<http://aif.cedram.org/item?id=AIF_2016__66_3_1279_0>
© Association des Annales de l’institut Fourier, 2016,
Certains droits réservés.
Cet article est mis à disposition selon les termes de la licence
CREATIVE COMMONS ATTRIBUTION – PAS DE MODIFICATION 3.0 FRANCE.
http://creativecommons.org/licenses/by-nd/3.0/fr/
L’accès aux articles de la revue « Annales de l’institut Fourier »
(http://aif.cedram.org/), implique l’accord avec les conditions générales
d’utilisation (http://aif.cedram.org/legal/).
cedram
Article mis en ligne dans le cadre du
Centre de diffusion des revues académiques de mathématiques
http://www.cedram.org/
Ann. Inst. Fourier, Grenoble
66, 3 (2016) 1279-1284
CORRIGENDUM TO “HOMOLOGY OF ORIGAMIS
WITH SYMMETRIES”
Annales de l’institut Fourier, vol. 64 (2014), n°3, 1131–1176
by Carlos MATHEUS,
Jean-Christophe YOCCOZ & David ZMIAIKOU
As it was kindly pointed out to us by Simion Filip, in Proposition 3.18
of [1] we gave a wrong description of the group Sp(Wa) when a is quater-
nionic.
More precisely, all facts claimed in [1] during the proof of this proposition
(from pages 1149 to 1153) are correct except for the penultimate phrase of
the argument:
“Moreover A preserves the symplectic form on Wa iff ι−1 ◦
A ◦ ι preserves the hermitian form
p∑
1
〈vm, v′m〉 −
p+q∑
p+1
〈vm, v′m〉
on V `a .”
As we are going to see in Section 1 below, after correcting the error
pointed out above, one has that, when a is quaternionic, the group Sp(Wa)
is not isomorphic to the unitary group UH(p, q) (as it was previously
claimed in [1]), but rather to the group O(`a,H) of matrices (with co-
efficients in H) satisfying A]A = I, where A] is the transpose matrix of
σ(A) and σ(a+ bi+ cj + dk) = a− bi+ cj + dk is a reversion on H.
In particular, besides the discussion of Sp(Wa) when a is quaternionic
undertaken in pages 1149 to 1153 in [1], all statements in [1] which were
based on the description of Sp(Wa) in Proposition 3.18 in [1] must be
changed. More concretely:
1280 Carlos MATHEUS, Jean-Christophe YOCCOZ & David ZMIAIKOU
— at page 1135, lines 6 and 7, the phrase “a quaternionic unitary
group UH(p, q) in the quaternionic case” must be replaced by “a
quaternionic orthogonal group O(`,H) in the quaternionic case”;
— at page 1135, the second item of Theorem 1.4 must be replaced by
“When Va is complex, resp. quaternionic, and Sp(Wa) is isomor-
phic to an unitary group UC(p, q), resp. O(`,H) with ` odd, the
multiplicity in Wa of the exponent 0 is at least |q − p|dimR(Va),
resp. dimR(Va);
— at page 1158, lines −5 to −3 must be replaced by: “Assume that
a is complex, resp. quaternionic. From Propositions 3.17, resp. the
new version of Proposition 3.18, there exists nonnegative integers
p, q with p + q = `a such that Sp(Wa) is isomorphic to UC(p, q),
resp. Sp(Wa) is isomorphic to O(`a,H).”;
— at page 1158, the statement of Proposition 4.11 must be replaced by
“If a is complex, the multiplicity of the exponent 0 in Wa is at least
|q− p|dimR(Va). If a is quaternionic and `a is odd, the multiplicity
of the exponent 0 in Wa is at least dimR(Va).”;
— at page 1159, the proof of Proposition 4.11 remains unchanged in
the complex case, but in the case of a quaternionic and `a odd,
the new argument is the following. “By Corollary 4.10, the list of
Lyapunov exponents in Wa has the form θ1 > · · · > θ`a where each
θm appears with multiplicity dimR(Va). By symplecticity, the list
θ1 > · · · > θ`a is symmetric with respect to 0. Therefore, θ(`a+1)/2 =
0 when `a is odd.”;
— at page 1173, the end of the phrase in Remark 5.13 must be
simply “. . . isotypical components of complex type” (instead of
“. . . isotypical components of complex or quaternionic type”) be-
cause there is no need for discussing the signature of Hermitian
form in the quaternionic case.
After these preliminaries, we complete this note by explaining how to
change the statement and proof of Proposition 3.18 in [1] to get a correct
description of Sp(Wa) when a is quaternionic.
1. Corrections to the statement and proof of
Proposition 3.18 in [1]
For the sake of convenience of the reader, instead of just giving a list
of punctual changes, we indicated below how to completely rewrite the
ANNALES DE L’INSTITUT FOURIER
CORRIGENDUM TO “HOMOLOGY OF ORIGAMIS WITH SYMMETRIES”1281
content of pages 1149 to 1153 in [1] in order to obtain a correct description
of Sp(Wa) when a is quaternionic.
* *
*
Consider finally the case where a is quaternionic. We fix an isomorphism
between Da and H. We equip Va with the structure of a right vector space
over H by setting
vz = z̄v, z ∈ H, v ∈ Va.
Here, z̄ = a− bi− cj − dk is the conjugate of the quaternion z = a+ bi+
cj + dk.
Recall that an hermitian form on a right vector space V over H is a map
H : V × V → H which satisfies
H(v, v1z1 + v2z2) = H(v, v1)z1 +H(v, v2)z2, ∀ v, v1, v2 ∈ V, z1, z2 ∈ H,
H(v, v′) = H(v′, v), ∀ v, v′ ∈ V.
Writing H = H0 +Hii+Hjj+Hkk, the R-bilinear form H0 is symmetric,
and the R-bilinear forms Hi, Hj , Hk are alternate. They are related by
H0(v, v′) = Hi(v, v′i) = Hj(v, v′j) = Hk(v, v′k).
For v, v′ ∈ V , z ∈ H, we have H(vz, v′z) = z̄H(v, v′)z, hence
Hi(vi, v′i) = Hi(v, v′),
Hi(vj, v′j) = −Hi(v, v′),
Hi(vk, v′k) = −Hi(v, v′).
Equivalently, one has, for v, v′ ∈ V, z ∈ H
Hi(v, v′z) = Hi(vσ(z), v′),
where σ is the reversion σ(a+ bi+ cj + dk) = a− bi+ cj + dk.
Conversely, if a R-bilinear alternate form Hi satisfy these relations, one
defines an hermitian form by
H(v, v′) = Hi(v, v′i) +Hi(v, v′)i+Hi(v, v′k)j −Hi(v, v′j)k.
On the irreducible Aut(M)-module Va, there exists, up to a positive
real scalar, a unique positive definite Aut(M)-invariant hermitian form 〈, 〉
(obtained as usual by averaging over the group an arbitrary positive definite
hermitian form). The space of Aut(M)-invariant R-bilinear forms on Va is
4-dimensional, generated by the four components of 〈 , 〉.
TOME 66 (2016), FASCICULE 3
1282 Carlos MATHEUS, Jean-Christophe YOCCOZ & David ZMIAIKOU
Proposition 3.18. — There exists an isomorphism of Aut(M)-modules
ι : V `aa → Wa such that the bilinear form {ι(v1, . . . , v`a), ι(v′1, . . . , v′`a)} is
the i-component of the hermitian form
∑`a
1 〈vm, v′m〉.
An element of Sp(Wa) is of the form ι−1◦A◦ι(v1, . . . , v`a) = (V1, . . . , V`a)
with Vm =
∑
n vnan,m , an,m ∈ H. The map A 7→ (am,n) is an isomorphism
from Sp(Wa) onto the group O(`a,H) of matrices satisfying A]A = I,
where A] is the transpose matrix of σ(A).
The proof of this proposition relies on the following three lemmas:
Lemma 3.19. — Let G be a finite group, and let W be an isotypic
G-module of quaternionic type. Let B be an alternate non-degenerate
G-invariant bilinear form onW . Then, there exists a non-zero vector v ∈W
and g ∈ G such that B(v, g.v) 6= 0.
Proof. — Assume that the conclusion of the lemma does not hold. Then,
one has B(v, g.v′) +B(v′, g.v) = 0 for all v, v′ ∈W . As B is alternate and
G-invariant, one has B(v, g2.v′) = B(v, v′) for all v, v′ ∈ W, g ∈ G. As B
is non-degenerate, this implies that g2.v′ = v′ for all v′ ∈ W, g ∈ G. Thus
G acts through a group where all non trivial elements are of order 2. Such
a group is abelian and W cannot be quaternionic. 
Lemma 3.20. — Under the hypotheses of the lemma above, one can
write
W = V1 ⊕ . . .⊕ V`,
where V1, . . . , V` are irreducible G-modules which are orthogonal for B.
Proof. — This is an immediate induction on the multiplicity ` of W .
From the lemma above, one can find v ∈W such that the restriction of B
to the irreducible G-module V1 generated by v is nonzero. Because V1 is
irreducible and B is G-invariant, this restriction is non-degenerate. Then,
the B-orthogonal W ′ of V1 in W is G-invariant and satisfies W = V1⊕W ′.
We conclude by applying to W ′ the induction hypothesis. 
Lemma 3.21. — Let b be an alternate Aut(M)-invariant nonzero
R-bilinear form on Va. There exists u ∈ H∗ such that the form bu(v, v′) :=
b(vu, v′u) is the i-component Bi of 〈 , 〉.
Proof. — Any nonzero alternate Aut(M)-invariant R-bilinear form b on
Va is non-degenerate. This allows to define an adjoint map σb : H → H
through b(v, v′a) = b(vσb(a), v′) (the R-endomorphism σb(a) of Va be-
longs to H as it commutes with the action of Aut(M)). The map σb is a
R-linear involution satisfying σb(aa′) = σb(a′)σb(a) and σb(a) = a for
ANNALES DE L’INSTITUT FOURIER
CORRIGENDUM TO “HOMOLOGY OF ORIGAMIS WITH SYMMETRIES”1283
a ∈ R. Therefore σb preserves the set of quaternions a such that a2 = −1,
which is nothing else than the purely imaginary quaternions of norm 1.
Observe that, for the i-component Bi of the hermitian scalar product on
Va, one has that σBi is the reversion σ.
As any nonzero alternate Aut(M)-invariant R-bilinear form b on Va is
a linear combination of the imaginary components Bi, Bj , Bk of 〈 , 〉 and
thus can be deformed to Bi through nonzero alternate Aut(M)-invariant
R-bilinear forms, we conclude that:
— there exists a unique (up to sign) quaternion a0 of norm 1 such that
σb(a0) = −a0 and, moreover, a0 is purely imaginary;
— σb is the identity on a hyperplane H of H containing R and the
purely imaginary quaternions inH are those satisfying za0+a0z=0.
Next we relate, for ūu = 1, σbu to σb. For v, v′ ∈ Va, we have
bu(v, v′a) = b(vu, v′au)
= b(vu, v′uūau)
= b(vuσb(ūau), v′u)
= b(vuσb(ūau)ūu, v′u)
= bu(vuσb(ūau)ū, v′),
and thus σbu(a) = uσb(ūau)ū. Hence, σbu = σBi iff ūiu = ±a0. As a0 is
purely imaginary of norm 1, it is possible to choose u such that ūiu = a0.
Then we have σbu(i) = −i and therefore σbu = σ.
It follows that there exists c ∈ R∗ such that bu is the i-component of
c〈 , 〉. Replacing u by c−1/2u if c > 0, by |c|−1/2ju if c < 0, we obtain the
required conclusion. 
Proof of Proposition 3.18. — According to Lemma 3.20 above, one can
choose an isomorphism of Aut(M)-modules ι0 : V `aa → Wa such that the
symplectic form is written in a diagonal way as
{ι0(v1, . . . , v`a), ι0(v′1, . . . , v′`a)} =
`a∑
1
b(m)(vm, v′m),
for some alternate nonzero Aut(M)-invariant R-bilinear forms b(m) on Va.
According to Lemma 3.21, one can find nonzero quaternions u1, . . . , u`a
such that, setting ι(v1, . . . , v`a) = ι0(v1u1, . . . , v`au`a), the alternate form
{ι(v1, . . . , v`a), ι(v′1, . . . , v′`a)}
is the i-component
∑`a
1 Bi(vm, v′m) of the hermitian form
∑`a
1 〈vm, v′m〉,
which proves the first assertion of the proposition. For the second assertion,
TOME 66 (2016), FASCICULE 3
1284 Carlos MATHEUS, Jean-Christophe YOCCOZ & David ZMIAIKOU
any automorphism A of the Aut(M)-module Wa is of the form ι−1 ◦ A ◦
ι(v1, . . . , v`a) = (V1, . . . , V`a) with Vm =
∑
n vnan,m , an,m ∈ H. Moreover
we have ∑
m
Bi(Vm, V ′m) =
∑
m
∑
n
∑
n′
Bi(vnan,m, v′n′an′,m)
=
∑
n
∑
n′
Bi(vn
∑
m
an,mσ(an′,m), v′n′)
This is equal to
∑
m Bi(vm, v′m) for all vm, v′m iff one has, for all n, n′∑
m
an,mσ(an′,m) = δn,n′ .
This means that the matrix associated to A satisfies AA] = I. 
BIBLIOGRAPHY
[1] C. Matheus, J.-C. Yoccoz & D. Zmiaikou, “Homology of origamis with symme-
tries”, Ann. Inst. Fourier (Grenoble) 64 (2014), no. 3, p. 1131-1176.
Manuscrit reçu le 14 mars 2015,
révisé le 18 mars 2016,
accepté le 29 avril 2016.
Carlos MATHEUS
Université Paris 13
Sorbonne Paris Cité
LAGA, CNRS (UMR 7539)
93430, Villetaneuse (France)
matheus.cmss@gmail.com
Jean-Christophe YOCCOZ
Collège de France (PSL)
3 rue d’Ulm
75005 Paris (France)
jean-c.yoccoz@college-de-france.fr
David ZMIAIKOU
Sanger Institute, Genome Campus
Hinxton
Cambridge CB10 1HH (UK)
david.zmiaikou@gmail.com
ANNALES DE L’INSTITUT FOURIER


Corrigendum to “Homology of origamis with symmetries”

Bichon, Julien

Neshveyev, Sergey

Yamashita, Makoto

Graded twisting of categories and quantum groups by group actions

Landesman, Aaron

Ruhm, Peter

Zhang, Robin

Spin canonical rings of log stacky curves

Lafforgue, Laurent

Du transfert automorphe de Langlands aux formules de Poisson non linéaires

Nicolas, Jean-Philippe

Conformal scattering on the Schwarzschild metric

Avila, Artur

Candela, Pablo

Towers for commuting endomorphisms, and combinatorial applications

Nguyen, Hong Duc

Invariants of plane curve singularities and Plücker formulas in positive characteristic

Gournay, Antoine

The Liouville property and Hilbertian compression

Araújo dos Santos, Raimundo

Hohlenwerger, Maria A.B.

Saeki, Osamu

Souza, Taciana O.

New examples of Neuwirth–Stallings pairs and non-trivial real Milnor fibrations

Bestvina, Mladen

Bromberg, Ken

Fujiwara, Koji

Stable commutator length on mapping class groups

Not a member yet