Construction of systems of differential equations of Okubo normal form with rigid monodromy

Abstract

Key words Rigid local system, system of Okubo normal form, monodromy representation MSC (2000) 34M35, 15A30For systems of differential equations of the form (xIn − T )dy/dx = Ay (systems of Okubo normal form), where A is an n × n constant matrix and T is an n × n constant diagonal matrix, two kinds of operations (extension and restriction) are defined. It is shown that every irreducible system of Okubo normal form of semi-simple type whose monodromy representation is rigid is obtained from a rank 1 system of Okubo normal form by a finite iteration of the operations. Moreover, an algorithm to calculate the generators of monodromy groups for rigid systems of Okubo normal form is given.

Katz's middle convolution and Yokoyama's extending operation

“…For example the condition ker(A j −λ j,ν )∩ ν =j ker A ν = {0} with dim ν =j ker A j = n j assures (3.23). (ii) Yokoyama [16] defines the extending operation for generic parameters λ j,ν , µ ν , ρ 1 and ρ 2 (cf. §1).…”

“…These operations do not change their indices of rigidity. (ii) The system (1.6) is called strongly reducible by [16] if there exists a non-trivial proper subspace of C n which is invariant under T and A. It is shown there that if the system is not strongly reducible, this property is kept by these operations.…”

“…Yokoyama [16] introduces an extension and a restriction of this system when A satisfies a certain genericity condition. Suppose according to the partition, namely A ij ∈ M (n i , n j , C).…”

“…Yokoyama [16] defines extensions (T ,Â) = E (T, A) for = 0, 1 and 2 with respect to two distinct complex numbers ρ 1 and ρ 2 when T , A and A ii (i = 1, . .…”

“…Yokoyama [16] proves that any irreducible rigid system of ONF with generic spectral parameters λ i,j and µ k is reduced to a rank 1 system by a finite iteration of the extensions and restrictions and obtained the monodromy of the system. In this note we clarify the direct relation between Yokoyama's operations and Katz's operations and then relax the assumption to define Yokoyama's operations (cf.…”