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Andrea Brini & Karoline van Gemst
Corrigendum: Mirror symmetry for extended affine Weyl groups
Tome 10 (2023), p. 1245-1246.
https://doi.org/10.5802/jep.243
© Les auteurs, 2023.
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e-ISSN : 2270-518X
Tome 10, 2023, p. 1245–1246 DOI: 10.5802/jep.243
CORRIGENDUM: MIRROR SYMMETRY FOR
EXTENDED AFFINE WEYL GROUPS
by Andrea Brini & Karoline van Gemst
Abstract. — We correct an error in [1]. There was a typo in the exponent of one of the factors
appearing in the expression of the Lyashko–Looijenga degrees in Corollary 5.2; the resulting
wrong formula was then used in Table 5.1 to tabulate these degrees for all Dynkin types.
We provide here the correct expressions for both Corollary 5.2 and Table 5.1.
Résumé (Corrigendum : Symétrie miroir pour les groupes de Weyl affines étendus)
Nous corrigeons une erreur dans [1]. Il y a une coquille dans l’exposant de l’un des facteurs
apparaissant dans l’expression des degrés de Lyashko–Looijenga au Corollaire 5.2 ; la formule
erronée qui en résulte est utilisée dans le tableau 5.1 pour lister ces degrés pour tous les types
de Dynkin. Nous indiquons ici les expressions correctes pour le corollaire 5.2 et le tableau 5.1.
The formula in the statement of [1, Cor. 5.2] contains a computational error,
whereby the exponent of (ωk, ωk) in the numerator should have been ℓR + 1 (and
not ℓR). We give the rectified statement below.
Corollary 5.2. — The degree of the LL-map of the Hurwitz stratum MLGω is
(ℓR + 1)!(ωk, ωk)
ℓR+1∏ℓR
j=1(ωj , ωk)
.
The incorrect formula was used to tabulate the topological degrees for all Dynkin
types other than Aℓ in Table 5.1. We correct those values in the table below.
Acknowledgements. — We became aware of an issue with the formula as presented
in [1, Cor. 5.2] after the appearance of [2]. We are very grateful to A. Takahashi for
correspondence related to this.
Mathematical subject classification (2020). — 53D45, 14B07, 20H15.
Keywords. — Frobenius manifolds, mirror symmetry, integrable systems.
This project has been supported by the Engineering and Physical Sciences Research Council under
grant agreement ref. EP/S003657/2.
e-ISSN: 2270-518X http://jep.centre-mersenne.org/
1246 A. Brini & K. van Gemst
Table 5.1. Lyashko–Looijenga degrees for all Dynkin types.
R gω nω dgω;nω ιω
(
MDZR
)
deg(LL)
Aℓ 0 (k − 1, ℓ− k) ℓ+ 1 H[µ]gω,nω
(ℓ+ 1− k)ℓ+1−k(k)kℓ!
(k − 1)!(ℓ− k)!
Bℓ 0 (ℓ− 2, ℓ− 2, 1) 2ℓ+ 1
(
H
[µ]
gω,nω
)µ2 2(ℓ+ 1)ℓ(ℓ− 1)ℓ
Cℓ 0 (ℓ− 1, ℓ− 1) 2ℓ
(
H
[µ]
gω,nω
)µ2 (ℓ+ 1)ℓℓ+1
Dℓ 0 (ℓ− 3, ℓ− 3, 1, 1) 2ℓ+ 2
(
H
[µ]
gω,nω
)µ2 4ℓ(ℓ2 − 1)(ℓ− 2)ℓ−1
E6 5 (5, 5, 2, 2, 2, 2, 2) 42 (A.1) 24 · 37 · 5 · 7
E7 33
(11, 5, 3, 11, 5,
3, 1, 1, 3, 3)
130 (A.2) 214 · 34 · 5 · 7
E8 128
(29, 29, 14, 14, 14,
14, 14, 14, 9, 9,
9, 9, 5, 5, 4,
4, 4, 4, 4, 4,
2, 2, 0, 0)
518 [6, 7] 25 · 36 · 56 · 7
F4 4 (5, 5, 2, 2, 2, 2) 36
(3.40);(
MLG[100000]E6
)µ2 24 · 34 · 5
G2 0 (1, 1, 1) 7
(3.41);(
MLG[1000]D4
)S3 23 · 32
References
[1] A. Brini & K. van Gemst – “Mirror symmetry for extended affine Weyl groups”, J. Éc. polytech.
Math. 9 (2022), p. 907–957.
[2] T. Otani, Y. Shiraishi & A. Takahashi – “The number of full exceptional collections modulo spher-
ical twists for extended Dynkin quivers”, 2023, arXiv:2308.04031.
Manuscript received 19th September 2023
accepted 28th September 2023
Andrea Brini, School of Mathematics and Statistics, University of Sheffield
S3 7RH, Sheffield, United Kingdom
On leave from CNRS, DR 13, Montpellier, France
E-mail : a.brini@sheffield.ac.uk
Karoline van Gemst, School of Mathematics and Statistics, University of Sheffield
S3 7RH, Sheffield, United Kingdom
E-mail : kvangemst1@sheffield.ac.uk
J.É.P. — M., 2023, tome 10


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