Abstract
Source algebras of block ideals are one of the main subjects in the modular representation theory of finite groups. Especially their bimodule structures attract much interest. In this paper, we shall treat block ideals with defect groups isomorphic to extraspecial $p$-groups of order $p^3$ and exponent $p$. We shall first analyze bimodule structures of these block ideals; we shall give a direct sum decomposition. We shall then prove that the images of the transfer maps on the cohomology rings of defect groups defined by the source algebras coincide with the cohomology rings of the block ideals in concern.
Funding Statement
This work was supported by JSPS KAKENHI Grant Number JP19K03442. This work was also supported by the Research Institute for Mathematical Sciences, an International Joint Usage/Research Center at Kyoto University.
Acknowledgment
The author would like to thank the referee for kind and valuable suggestions and advice.
Citation
Hiroki SASAKI. "Source algebras and cohomology algebras of block ideals of finite groups with defect groups isomorphic to extraspecial $p$-groups." Hokkaido Math. J. 53 (3) 443 - 462, October 2024. https://doi.org/10.14492/hokmj/2023-715
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