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The Broué invariant of a Morita equivalence with an endopermutation source

  • Xin Huang
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Abstract

A perfect isometry I (introduced by Broué) between two blocks b and c is a frequent phenomenon in the block theory of finite groups. It maps an irreducible character ψ of c to ± an irreducible character of b. Broué proved that the ratio of the codegrees of ψ and I(ψ) is a rational number with p-value zero and that its class in 𝔽p is independent of ψ. This element is called the Broué invariant of I by Boltje. The goal of this paper is to show that if I comes from a Morita equivalence with an endopermutation source V, then, up to a sign, the Broué invariant of I is determined by local data of b and c. Therefore, up to a sign, it is independent of the endopermutation-source Morita equivalence. Moreover, we show that the sign factor is given by the reduction of the rank of V modulo p. As a corollary, we obtain that the Isaacs--Navarro refinement of the Alperin--McKay conjecture holds for inertial blocks.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026broué,
  title = {{The Broué invariant of a Morita equivalence with an endopermutation source}},
  author = {Xin Huang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.06560},
  eprint = {2610.06560},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.06560},
}
RIS
TY  - UNPB
TI  - The Broué invariant of a Morita equivalence with an endopermutation source
AU  - Xin Huang
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.06560
UR  - https://arxiv.org/abs/2610.06560
AB  - A perfect isometry I (introduced by Broué) between two blocks b and c is a frequent phenomenon in the block theory of finite groups. It maps an irreducible character ψ of c to ± an irreducible character of b. Broué proved that the ratio of the codegrees of ψ and I(ψ) is a rational number with p-value zero and that its class in 𝔽_p is independent of ψ. This element is called the Broué invariant of I by Boltje. The goal of this paper is to show that if I comes from a Morita equivalence with an endopermutation source V, then, up to a sign, the Broué invariant of I is determined by local data of b and c. Therefore, up to a sign, it is independent of the endopermutation-source Morita equivalence. Moreover, we show that the sign factor is given by the reduction of the rank of V modulo p. As a corollary, we obtain that the Isaacs--Navarro refinement of the Alperin--McKay conjecture holds for inertial blocks.
ER  -