Matrix discrepancy for representations of finite groups

Authors

Afonso S. Bandeira and Helmut Bölcskei

Reference

Applied and Computational Harmonic Analysis, June 2026, submitted.

Abstract

We prove the group version of the Matrix Spencer conjecture. For every finite group $G$, there exist signs $\varepsilon\in\{\pm1\}^G$ such that $$\left\| \sum_{g\in G} \varepsilon_g\rho(g) \right\|\leq C\, \sqrt{|G|},$$ where $\rho$ is the left regular representation of $G$ and $C$ is a universal constant. This conjecture was posed in [BKMZ24], which settled it for simple groups; we establish it for all finite groups, combining the Peter–Weyl decomposition with the intrinsic-freeness inequalities of [BBvH23] in an iterated partial-coloring argument.

Keywords

Matrix Spencer problem, finite groups, discrepancy theory

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