Generically constrained quantum isotropy

Abstract

Let V be a finite-dimensional unitary representation of a compact quantum group \mathrm{G} and denote by \mathrm{G}_{W} the isotropy subgroup of a linear subspace W\le V regarded as a point in the Grassmannian \mathbb{G}(V) . We show that the space of those W\in \mathbb{G}(V) for which \mathrm{G}_{W} acts trivially on W (or V ) is open in the Zariski topology of the Weil restriction \mathrm{Res}_{\mathbb{C}/\mathbb{R}}\mathbb{G}(V) . More generally, this holds for the space of W for which (a) the \mathrm{G}_{W} -action factors through its abelianization or (b) the summands of the \mathrm{G}_{W} -representation on W (or V ) are otherwise dimensionally constrained. The results generalize analogous classical generic rigidity statements useful in establishing the triviality of the classical automorphism groups of random quantum graphs in the matrix algebra M_{n} and can be put to similar use in fully non-commutative versions of those results (quantum graphs, quantum groups).