A perfect isometry (introduced by Broué) between two blocks and is a frequent phenomenon in the block theory of finite groups. It maps an irreducible character of to an irreducible character of . Broué proved that the ratio of the codegrees of and is a rational number with -value zero and that its class in is independent of . We call this element the Broué invariant of . The goal of this paper is to show that if comes from a -permutation equivalence or a splendid Rickard equivalence between and then, up to a sign, the Broué invariant of is determined by local data of and and therefore, up to a sign, is independent of the -permutation equivalence or splendid Rickard equivalence. Apart from results on -permutation equivalences, our proof requires new results on extended tensor products and bisets that are also proved in this paper. As application of the theorem on the Broué invariant we show that various refinements of the Alperin–McKay Conjecture, introduced by Isaacs–Navarro, Navarro, and Turull are consequences of -permutation equivalences or splendid Rickard equivalences over a sufficiently large complete discrete valuation ring or over , depending on the refinement.
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Keywords: Blocks of finite groups, perfect isometry, splendid Rickard equivalence, $p$-permutation equivalence, bisets
Boltje, Robert 1
@article{ART_2024__1_4_517_0, author = {Boltje, Robert}, title = {The {Brou\'e} invariant of a $p$-permutation equivalence}, journal = {Annals of Representation Theory}, pages = {517--537}, publisher = {The Publishers of ART}, volume = {1}, number = {4}, year = {2024}, doi = {10.5802/art.17}, language = {en}, url = {https://art.centre-mersenne.org/articles/10.5802/art.17/} }
TY - JOUR AU - Boltje, Robert TI - The Broué invariant of a $p$-permutation equivalence JO - Annals of Representation Theory PY - 2024 SP - 517 EP - 537 VL - 1 IS - 4 PB - The Publishers of ART UR - https://art.centre-mersenne.org/articles/10.5802/art.17/ DO - 10.5802/art.17 LA - en ID - ART_2024__1_4_517_0 ER -
Boltje, Robert. The Broué invariant of a $p$-permutation equivalence. Annals of Representation Theory, Volume 1 (2024) no. 4, pp. 517-537. doi : 10.5802/art.17. https://art.centre-mersenne.org/articles/10.5802/art.17/
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