Essential algebra of the shifted Burnside biset functor with abelian shift
Abstract
Let T be a finite group and let kBT denote the shifted Burnside biset functor over a commutative ring k. We study the essential algebra kBT̂(G) of kBT at an arbitrary finite group G. We first establish criteria determining when a covering subgroup of G× G× T factors, with respect to the star product, through a group of order strictly smaller than |G|; the first criterion places no hypothesis on T. As an application, we describe kBT̂(G) completely whenever no nontrivial quotient of G is isomorphic to a section of T, extending the coprime case treated by Romero. When T is abelian and |T| is invertible in k, we prove that kBT̂(G) decomposes as a direct sum of matrix algebras over the group algebras of the groups OutT(A), where A runs over the linkage classes of reduced subgroups of G× T, and we parametrize the simple modules of kBT̂(G).
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BibTeX
@misc{coskun2026essential,
title = {{Essential algebra of the shifted Burnside biset functor with abelian shift}},
author = {Olcay Coşkun and Ruslan Muslumov},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.08616},
eprint = {2609.08616},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.08616},
}
RIS
TY - UNPB
TI - Essential algebra of the shifted Burnside biset functor with abelian shift
AU - Olcay Coşkun
AU - Ruslan Muslumov
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2609.08616
UR - https://arxiv.org/abs/2609.08616
AB - Let T be a finite group and let kB_T denote the shifted Burnside biset functor over a commutative ring k. We study the essential algebra kB_T̂(G) of kB_T at an arbitrary finite group G. We first establish criteria determining when a covering subgroup of G× G× T factors, with respect to the star product, through a group of order strictly smaller than |G|; the first criterion places no hypothesis on T. As an application, we describe kB_T̂(G) completely whenever no nontrivial quotient of G is isomorphic to a section of T, extending the coprime case treated by Romero. When T is abelian and |T| is invertible in k, we prove that kB_T̂(G) decomposes as a direct sum of matrix algebras over the group algebras of the groups Out_T(A), where A runs over the linkage classes of reduced subgroups of G× T, and we parametrize the simple modules of kB_T̂(G).
ER -