Equivariant spaces of matrices of constant corank one
Ada Boralevi, Daniele Faenzi, Dragoş Frăţilă
TL;DR
This work classifies spaces of matrices that arise from irreducible representations of a reductive group $G$ and are equivariant under $G$, focusing on the constant corank $1$ case. The main theorem provides necessary and sufficient weight conditions: there exist a simple root index $i$ with $\lambda=\nu+\mu-\alpha_i$ and $\mu$ simultaneously a multiple of $\nu$ and of the fundamental weight $\omega_i$, ensuring constant corank $1$; the proof combines Frobenius reciprocity, weight-space analysis, and $\mathrm{SL}_2$-type arguments, with a constructive converse using $G$-equivariant bundle maps on $\mathbb{P}(V(\nu))$. The results recover known $\mathrm{SL}_2$ examples (and their $\mathrm{SL}_n$ analogues at $i=1$ from prior work) and provide a framework to examine related questions for wedge powers, PRV components, and vector-bundle geometry, including the Tango-type phenomena in the corank-one setting. Overall, the paper advances the systematic understanding of when equivariant matrix spaces have constant corank and connects representation-theoretic data to geometric and bundle-theoretic consequences.
Abstract
We study spaces of matrices coming from irreducible representations of reductive groups over an algebraically closed field of characteristic zero and we completely classify those of constant corank one. In particular, we recover the examples coming from symmetric forms discovered in [BFL22].