A Note on Group Representations, Determinantal Hypersurfaces and Their Quantizations

Abstract

Recently, there have been exciting developments on the interplay between representation theory of finite groups and determinantal hypersurfaces. For example, a finite Coxeter group is determined by the determinantal hypersurface described by its natural generators under the regular representation. This short note solves three problems about extending this result in the negative. On the affirmative side, it is shown that a quantization of a determinantal hypersurface, the so-called free locus, correlates well with representation theory. If \(A_1,\dots ,A_\ell \in \operatorname {\mathrm {GL}}_d(\mathbb {C})\) generate a finite group G, then the family of hypersurfaces \(\{X\in \operatorname {\mathrm {M}}_{n}(\mathbb {C})^d\colon \det (I+A_1\otimes X_1+\cdots +A_\ell \otimes X_\ell )=0 \}\) for \(n\in \mathbb {N}\) determines G up to isomorphism.