Principal 2-blocks with wreathed defect groups up to splendid Morita equivalence
Koshitani, Shigeo1; Lassueur, Caroline2; Sambale, Benjamin2
1 Chiba University Department of Mathematics and Informatics Graduate School of Science 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522 (Japan)
2 Leibniz Universität Hannover Institut für Algebra, Zahlentheorie und Diskrete Mathematik Welfengarten 1 D-30167 Hannover (Germany)
Annals of Representation Theory, Volume 1 (2024) no. 3, pp. 439-463.

We classify principal 2-blocks of finite groups G with Sylow 2-subgroups isomorphic to a wreathed 2-group C 2 n ≀C 2 with n≥2 up to Morita equivalence and up to splendid Morita equivalence. As a consequence, we obtain that Puig’s Finiteness Conjecture holds for such blocks. Furthermore, we obtain a classification of such groups modulo O 2 ′ (G), which is a purely group theoretical result and of independent interest. Methods previously applied to blocks of tame representation type are used. They are, however, further developed in order to deal with blocks of wild representation type.

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DOI: 10.5802/art.16
Classification: 20C05, 20C20, 20C15, 20C33, 16D90
Keywords: wreathed $2$-group, Morita equivalence, splendid Morita equivalence, Puig’s Finiteness Conjecture, principal block, trivial source module, $p$-permutation module, Scott module, Brauer indecomposability, decomposition matrix
Koshitani, Shigeo 1; Lassueur, Caroline 2; Sambale, Benjamin 2

1 Chiba University Department of Mathematics and Informatics Graduate School of Science 1-33 Yayoi-cho, Inage-ku, Chiba 263-8522 (Japan)
2 Leibniz Universität Hannover Institut für Algebra, Zahlentheorie und Diskrete Mathematik Welfengarten 1 D-30167 Hannover (Germany)
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Koshitani, Shigeo; Lassueur, Caroline; Sambale, Benjamin. Principal $2$-blocks with wreathed defect groups up to splendid Morita equivalence. Annals of Representation Theory, Volume 1 (2024) no. 3, pp. 439-463. doi : 10.5802/art.16. https://art.centre-mersenne.org/articles/10.5802/art.16/

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