Canonical decomposition of irreducible linear differential operators with symplectic or orthogonal differential Galois groups

Abstract

We first revisit an order-six linear differential operator, already introduced in a previous paper, having a solution which is a diagonal of a rational function of three variables. This linear differential operator is such that its exterior square has a rational solution, indicating that it has a selected differential Galois group, and is actually homomorphic to its adjoint. We obtain the two corresponding intertwiners giving this homomorphism to the adjoint. We show that these intertwiners are also homomorphic to their adjoint and have a simple decomposition, already underlined in a previous paper, in terms of order-two self-adjoint operators. From these results, we deduce a new form of decomposition of operators for this selected order-six linear differential operator in terms of three order-two self-adjoint operators. We generalize this decomposition to decomposition in terms of three self-adjoint operators of arbitrary orders, provided the three orders have the same parity. We then generalize the previous decomposition to decompositions in terms of an arbitrary number of self-adjoint operators of the same parity order. This yields an infinite family of linear differential operators homomorphic to their adjoint, and, thus, with a selected differential Galois group. We show that the equivalence of such operators, with selected differential Galois groups, is compatible with these canonical decompositions. The rational solutions of the symmetric, or exterior, squares of these selected operators are, noticeably, seen to depend only on the rightmost self-adjoint operator in the decomposition. These results, and tools, are applied on operators of large orders. For instance, it is seen that a large set of (quite massive) operators, associated with reflexive 4-polytopes defining Calabi–Yau three-folds, obtained recently by Lairez, correspond to a particular form of the decomposition detailed in this paper. All the results of this paper can be seen as providing an algebraic characterization of linear differential operators with selected symplectic or orthogonal differential Galois groups.