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For the symplectic Grassmannian $\text{SpG}(2,2n)$ of $2$-dimensionalisotropic subspaces in a $2n$-dimensional vector space over an algebraicallyclosed field of characteristic zero endowed with a symplectic form and with thenatural action of an $n$-dimensional torus $T$ on it, we characterize itsirreducible $T$-invariant subvarieties. This characterization is in terms ofsymplectic Coxeter matroids, and we use this result to give a completecharacterization of the symplectic matroids of rank $2$ which are representableover $\mathbb{C}$.

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