On $T$-invariant subvarieties of symplectic Grassmannians and representability of rank $2$ symplectic matroids over ${\mathbb C}$

TL;DR

This paper characterizes the irreducible T-invariant subvarieties of symplectic Grassmannians using symplectic Coxeter matroids and applies this to classify rank 2 symplectic matroids over the complex numbers.

Contribution

It provides a complete characterization of T-invariant subvarieties in symplectic Grassmannians and classifies rank 2 symplectic matroids over or the first time.

Findings

Irreducible T-invariant subvarieties correspond to symplectic Coxeter matroids.

Complete classification of rank 2 symplectic matroids over ound.

Connection established between geometric subvarieties and combinatorial matroid structures.

Abstract

For the symplectic Grassmannian $\text{SpG}(2,2n)$ of $2$-dimensional isotropic subspaces in a $2n$-dimensional vector space over an algebraically closed field of characteristic zero endowed with a symplectic form and with the natural action of an $n$-dimensional torus $T$ on it, we characterize its irreducible $T$-invariant subvarieties. This characterization is in terms of symplectic Coxeter matroids, and we use this result to give a complete characterization of the symplectic matroids of rank $2$ which are representable over $\mathbb{C}$.