Abstract
Let R be a fusion ring and Rℂ:= R ⊗ℤ ℂ be the corresponding fusion algebra. We first show that the algebra Rℂ has only one left (right, two-sided) cell and the corresponding left (right, two-sided) cell module. Then we prove that, up to isomorphism, Rℂ admits a unique special module, which is 1-dimensional and given by the Frobenius-Perron homomorphism FPdim. Moreover, as an example, we explicitly determine the special module of the interpolated fusion algebra R(PSL(2, q)):= r(PSL(2, q)) ⊗ℤ ℂ up to isomorphism, where r(PSL(2, q)) is the interpolated fusion ring with even q ⩾ 2.
This is a preview of subscription content, log in via an institution to check access.
References
L. F. Cao, H. X. Chen, L. B. Li: The cell modules of the Green algebra of Drinfel’d quantum double D(H4). Acta Math. Sin., Engl. Ser. 38 (2022), 1116–1132.
P. Etingof, S. Gelaki, D. Nikshych, V. Ostrik: Tensor Categories. Mathematical Surveys and Monographs 205. AMS, Providence, 2015.
G. Frobenius: Über Matrizen aus positiven Elementen. Berl. Ber. 1908 (1908), 471–476. (In German.)
G. Frobenius: Über Matrizen aus positiven Elementen. II. Berl. Ber. 1909 (1909), 514–518. (In German.)
F. R. Gantmacher: The Theory of Matrices. Vol. 1. AMS Chelsea Publishing, Providence, 1998.
D. Kazhdan, G. Lusztig: Representations of Coxeter groups and Hecke algebras. Invent. Math. 53 (1979), 165–184.
K. Kildetoft, V. Mazorchuk: Special modules over positively based algebras. Doc. Math. 21 (2016), 1171–1192.
G. Kudryavtseva, V. Mazorchuk: On multisemigroups. Port. Math. (N.S.) 72 (2015), 47–80.
S. Lin, S. Yang: Representations of a class of positively based algebras. Czech. Math. J. 73 (2023), 811–838.
Z. Liu, S. Palcoux, Y. Ren: Interpolated family of non group-like simple integral fusion rings of Lie type. Available at https://arxiv.org/abs/2102.01663 (2021), 29 pages.
M. Lorenz: Some applications of Frobenius algebras to Hopf algebras. Groups, Algebras and Applications. Contemporary Mathematics 537. AMS, Providence, 2011, pp. 269–289.
G. Lusztig: Leading coefficients of character values of Hecke algebras. Representations of Finite Groups. Proceedings of Symposia in Pure Mathematics 47. AMS, Providence, 1987, pp. 235–262.
V. Mazorchuk, V. Miemietz: Cell 2-representations of finitary 2-categories. Compos. Math. 147 (2011), 1519–1545.
O. Perron: Zur Theorie der Matrices. Math. Ann. 64 (1907), 248–263. (In German.)
Acknowledgement
The authors are grateful to V. Mazorchuk for sending the references [7], [8], [13] and also thank the anonymous referee for numerous suggestions that helped to improve the paper substantially.
Author information
Authors and Affiliations
Corresponding author
Additional information
The first author was supported by Scientific Research and Innovation Project of Graduate Students in Jiangsu Province (Grant No. KYCX22-3448) and National Natural Science Foundation of China (Grant No. 12371041). The second author was supported by the National Natural Science Foundation of China (Grant No. 12071412).
Rights and permissions
About this article
Cite this article
Cao, L., Chen, H. Special modules for R(PSL(2, q)). Czech Math J 73, 1301–1317 (2023). https://doi.org/10.21136/CMJ.2023.0002-23
Received:
Published:
Issue Date:
DOI: https://doi.org/10.21136/CMJ.2023.0002-23