Matrix Discrepancy for Representations of Finite Groups

Abstract

We prove the group version of the Matrix Spencer conjecture. For every finite group $G$, there exist signs $\epsilon \in \{\pm 1{\}}^G$ such that $$\Vert \sum _{g\in G}{\epsilon }_g\ddot{I}(g)\Vert \le C\sqrt{|G|},$$ where $\ddot{I}$ 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.

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