Blockwise Alperin weight conjecture

About 10 years old · traced to 2016

Let $G$ be a finite group, $p$ a prime, and $B$ a $p$ -block of $G$ . A $p$ -weight of $B$ is a $p$ -weight belonging to the block $B$ , and the irreducible $p$ -Brauer characters of $B$ are those belonging to $B$ . Blockwise Alperin weight conjecture. The number of irreducible $p$ -Brauer characters of $B$ equals the number of $G$ -conjugacy classes of $p$ -weights of $B$ . This is a refinement of Alperin's weight conjecture that counts the relevant modular characters and local weights block by block; it remains open in general.

References

Primary source

Julian Brough and Britta Späth, “A criterion for the inductive Alperin weight condition”, arXiv:2009.02074 (2020).

Additional references

2 papers in this index state this conjecture (2016–2020). The statement above is taken from the most recent of them; the others are arXiv:1603.05065.

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for every finite group, every prime p, and every p-block, the number of irreducible Brauer characters equals the number of conjugacy classes of block weights; this is the numerical blockwise Alperin weight conjecture.

Repository: https://github.com/openai/math

Manuscript: https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-Blockwise-Alperin-Weight-Conjecture-September-23-2026/paper.pdf

OpenAI-202-01-The-Blockwise-Alperin-Weight-Conjecture.pdf791,829 bytesOpen