Blocky Matrices and Group Idempotents
Abstract
We prove a common generalization of two structure theorems: the dimension-free decomposition theorem for idempotent Schur multipliers and the idempotent theorem in harmonic analysis. Roughly speaking, our result shows that an invariant integer-valued kernel with Hilbert-space factorization norm γ admits a signed decomposition into at most 2O(γ4) elementary pieces. In the matrix setting these pieces are blocky matrices, while in the group setting they are indicators of cosets. In the locally compact abelian setting this quantitatively strengthens the theorem of Green and Sanders, while in the non-abelian setting it gives a quantitative strengthening of Host's idempotent theorem and, for finite groups, of Sanders's quantitative result. It also improves the exponent in the dimension-free matrix decomposition from γ6 to γ4.
Related
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- Unique direct-sum decompositions of infinite-dimensional unitary representations of the infinite dihedral group via real CMV matrices
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- Nonlinear Lie Triple Centralizers of Unital Algebras with Idempotents
BibTeX
@misc{carenini2026blocky,
title = {{Blocky Matrices and Group Idempotents}},
author = {Gaia Carenini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27951},
eprint = {2609.27951},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27951},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Blocky Matrices and Group Idempotents
AU - Gaia Carenini
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27951
UR - https://arxiv.org/abs/2609.27951
AB - We prove a common generalization of two structure theorems: the dimension-free decomposition theorem for idempotent Schur multipliers and the idempotent theorem in harmonic analysis. Roughly speaking, our result shows that an invariant integer-valued kernel with Hilbert-space factorization norm γ admits a signed decomposition into at most 2^O(γ^4) elementary pieces. In the matrix setting these pieces are blocky matrices, while in the group setting they are indicators of cosets. In the locally compact abelian setting this quantitatively strengthens the theorem of Green and Sanders, while in the non-abelian setting it gives a quantitative strengthening of Host's idempotent theorem and, for finite groups, of Sanders's quantitative result. It also improves the exponent in the dimension-free matrix decomposition from γ^6 to γ^4.
ER -