Canonical decomposition of linear differential operators with selected differential Galois groups

Abstract

We revisit an order-six linear differential operator having a solution which is a diagonal of a rational function of three variables. 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 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 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 3-folds, obtained recently by P. Lairez, correspond to a particular form of the decomposition detailed in this paper.

arXiv comment40 pages

References in corpus (14)

• Lattice Green functions in all dimensions
• Experimental mathematics on the magnetic susceptibility of the square lattice Ising model
• The Fuchsian differential equation of the square lattice Ising model $\chi (3)$ susceptibility
• Ising n-fold integrals as diagonals of rational functions and integrality of series expansions
• Square lattice Ising model susceptibility: connection matrices and singular behavior of ${\chi }^{(3)}$ and ${\chi }^{(4)}$
• The diagonal Ising susceptibility
• The Ising model: from elliptic curves to modular forms and Calabi-Yau equations
• Singularities of $n$-fold integrals of the Ising class and the theory of elliptic curves
• Lattice Green's Functions of the Higher-Dimensional Face-Centered Cubic Lattices
• High order Fuchsian equations for the square lattice Ising model: ${\chi }^{(6)}$
• Lattice Green Function (at 0) for the 4d Hypercubic Lattice
• Landau singularities and singularities of holonomic integrals of the Ising class
• Holonomic functions of several complex variables and singularities of anisotropic Ising n-fold integrals
• Deformations of Fuchsian Systems of Linear Differential Equations and the Schlesinger System

Cited by in corpus (4)

• Diagonals of rational functions and selected differential Galois groups
• Lattice Green Functions: the d-dimensional face-centred cubic lattice, d=8, 9, 10, 11, 12
• Scaling functions in the square Ising model
• Schwarzian conditions for linear differential operators with selected differential Galois groups (unabridged version)