Fusion-stable endosplit p-permutation resolutions

@misc{huang2026fusionstable,
  title = {{Fusion-stable endosplit $p$-permutation resolutions}},
  author = {Xin Huang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.15971},
  eprint = {2609.15971},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.15971},
}

TY  - UNPB
TI  - Fusion-stable endosplit p-permutation resolutions
AU  - Xin Huang
PY  - 2026
DA  - 2026/09/16
PB  - arXiv
DO  - 10.48550/arxiv.2609.15971
UR  - https://arxiv.org/abs/2609.15971
AB  - Let k be a field of characteristic p>0, ℱ a saturated fusion system over a finite p-group P, and V an indecomposable capped endopermutation kP-module. Let D_k^Ω(P) be the subgroup of the Dade group D_k(P) generated by all the relative syzygies. It is known that V has an endosplit p-permutation resolution if and only if the Dade class [V] belongs to D_k^Ω(P). We show that the resolution can be chosen to be ℱ-stable if and only if V is ℱ-stable. As an application, we prove the following folklore result: if two blocks of finite groups are Morita equivalent via a bimodule with an endopermutation kP-source V such that [V]∈ D_k^Ω(P), then these two blocks are splendidly Rickard equivalent.
ER  -