A classification of group gradings on incidence algebras over commutative rings
Abstract
Let R be a commutative ring with 1, P a locally finite partially ordered set, and G a group. We derive necessary and sufficient conditions for an R-algebra isomorphism between the incidence algebra I(P,R) and the group algebra RG. Then, for an indecomposable ring R, a finite poset P and an arbitrary group G, we classify the G-gradings of I(P,R) up to graded isomorphism. The classification rests on a complete set of primitive orthogonal homogeneous idempotents. The corner algebras are split group algebras of finite abelian subgroups of G, and the off-diagonal Peirce blocks are multiplicity-free sums of bimodules induced from characters of double coset stabilizers. Graded isomorphisms are shown to have a rigid form, and a grading is determined up to graded isomorphism by the poset of idempotents, the corner groups, the types of the atomic bimodules and the structure constants of their multiplication. The data which occur are characterized by polynomial conditions, and over an algebraically closed field of characteristic zero only finitely many graded isomorphism classes share given partial invariants. An example shows that the structure constants cannot be omitted. Some previous results are extended and enhanced, while providing alternative proofs for some known facts.
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BibTeX
@misc{hindi2026classification,
title = {{A classification of group gradings on incidence algebras over commutative rings}},
author = {Felipe de Mattos Chafik Hindi and Waldeck Schutzer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21184},
eprint = {2609.21184},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21184},
}
RIS
TY - UNPB
TI - A classification of group gradings on incidence algebras over commutative rings
AU - Felipe de Mattos Chafik Hindi
AU - Waldeck Schutzer
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21184
UR - https://arxiv.org/abs/2609.21184
AB - Let R be a commutative ring with 1, P a locally finite partially ordered set, and G a group. We derive necessary and sufficient conditions for an R-algebra isomorphism between the incidence algebra I(P,R) and the group algebra RG. Then, for an indecomposable ring R, a finite poset P and an arbitrary group G, we classify the G-gradings of I(P,R) up to graded isomorphism. The classification rests on a complete set of primitive orthogonal homogeneous idempotents. The corner algebras are split group algebras of finite abelian subgroups of G, and the off-diagonal Peirce blocks are multiplicity-free sums of bimodules induced from characters of double coset stabilizers. Graded isomorphisms are shown to have a rigid form, and a grading is determined up to graded isomorphism by the poset of idempotents, the corner groups, the types of the atomic bimodules and the structure constants of their multiplication. The data which occur are characterized by polynomial conditions, and over an algebraically closed field of characteristic zero only finitely many graded isomorphism classes share given partial invariants. An example shows that the structure constants cannot be omitted. Some previous results are extended and enhanced, while providing alternative proofs for some known facts.
ER -