Elements of the Green ring related to Alperin's weight conjecture
Abstract
J. L. Alperin introduced his celebrated weight conjecture, a major question in the modular representation theory of finite groups, which has now been studied for several decades. Alperin viewed this conjecture as the beginning of a ``Lie theory for the general finite group", as the title of the article indicates. This paper is intended as a tribute to Alperin's insight. In this note, we place an emphasis on projective modules and putting the numerical questions arising from the weight conjecture into a ``module form" within the appropriate Green ring of RG, when G is a finite group, and R is a suitable complete discrete valuation ring with residue field of prime characteristic p. We believe that this work moves in the direction of a structural approach to Alperin's weight conjecture.
Related
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BibTeX
@misc{robinson2026elements,
title = {{Elements of the Green ring related to Alperin's weight conjecture}},
author = {Geoffrey R. Robinson},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27988},
eprint = {2610.27988},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27988},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Elements of the Green ring related to Alperin's weight conjecture
AU - Geoffrey R. Robinson
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.27988
UR - https://arxiv.org/abs/2610.27988
AB - J. L. Alperin introduced his celebrated weight conjecture, a major question in the modular representation theory of finite groups, which has now been studied for several decades. Alperin viewed this conjecture as the beginning of a ``Lie theory for the general finite group", as the title of the article indicates. This paper is intended as a tribute to Alperin's insight. In this note, we place an emphasis on projective modules and putting the numerical questions arising from the weight conjecture into a ``module form" within the appropriate Green ring of RG, when G is a finite group, and R is a suitable complete discrete valuation ring with residue field of prime characteristic p. We believe that this work moves in the direction of a structural approach to Alperin's weight conjecture.
ER -