Suhas Vadan Gondi | Searcharxiv

arXiv2026

Border rank lower bounds for families of GL(V)-invariant tensors

We give non-trivial lower bounds for the border rank of families of $\mathbf{GL}(V)$-invariant tensors in $U\otimes \mathbf{S}_λV\otimes \mathbf{S}_μV$ where $U$ is $V$, $\mathrm{Sym}^2V$ or $\bigwedge^2V$. In particular, we provide a family of tensors with border rank reaching arbitrarily close to $2\ell$ in the unbalanced case, where $\ell$ is the largest ambient vector space dimension. We do this by resolving a conjecture introduced by Wu, and obtaining new results on $6j$-symbols as a byproduct. We then generalize our results to $\mathrm{Sym}^2V$ and $\bigwedge^2 V$ using novel techniques based on an application of a theorem of Kostant and Kempf collapsing.

math.AG↗

arXiv2026

Syzygies of Isotropic Kalman Varieties

Let $L$ be a subspace of a complex vector space $V$ and fix $s \leq \dim{L}$. The (type A) Kalman variety consists of all endomorphisms of $V$ that have an $s$-dimensional invariant subspace in $L$. We introduce a generalization where $V$ and $L$ are symplectic vector spaces. We fix an isotropic subspace $W \subseteq V$ satisfying $W^\perp = W \oplus L$. The isotropic (type C) Kalman variety consists of symplectic morphisms of $V$ that have an invariant coisotropic subspace of a prescribed dimension inside $W^\perp$. We are mainly interested in studying the Lagrangian case. In type C, we prove analogues of results known for type A Kalman varieties; in particular, we determine the defining equations, compute geometric invariants, and analyze their singularities. We conjecture the existence of a long exact sequence relating the structure sheaves. Based on the results in the symplectic case, we describe Kalman variety analogues with respect to endomorphisms of odd orthogonal (type B) and even orthogonal (type D) vector spaces.

math.AG↗