Invariant theory for linear differential systems modeled after the Grassmannian {${\rm Gr}(n,2n)$}

Abstract

We find invariants for the differential systems of rank $2n$ in $n^{2}$ variables with $n$ unknowns under the linear changes of the unknowns with variable coefficients. We look for a set of coefficients that determines the other coefficients, and give transformation rules under the linear changes above and coordinate changes. These can be considered as a generalization of the Schwarzian derivative, which is the invariant for second order ordinary differential equations under the change of the unknown by multiplying a non-zero function. Special treatment is done when $n = 2$: the conformal structure obtained through the Plücker embedding is studied, and a relation with line congruences is discussed.