A differential equation with monodromy group 2.J2

Abstract

A DIFFERENTIAL EQUATION WITH MONODROMY GROUP 2.J 2 STEFAN REITERAbstract. We construct a sixth order differential equation having the central extension of C 2 by the Hall-Janko group J 2 as monodromy group. Moreover it arises from an iterated application of tensor products and convolution operations from a first order differential equation. IntroductionAccording to [13, 5.6.1] there are two important constructions of the sporadic simple Hall-Janko group J 2 of order 604800, namely as permutation group on 100 points by Marshall Hall and as a quaternionic reflection group in 3 dimensions in connection with the Leech lattice. It is well known that 2.J 2 , the central extension of C 2 by the HallJanko group J 2 , is an irreducible subgroup of Sp 6 (C) [3, p. 42-43]. Generators of the six dimensional representation of 2.J 2 over Q(ζ 5 ) were already determined by Lindsey II [6]. Here we show that the group 2.J 2 appears as a monodromy group of a sixth order differential equation that can be constructed by an iterated application of tensor products and convolution operations from a first order differential equation. Throughout the article, letGalois group of L, and hence the monodromy group of L, is contained in the symplectic group Sp n (C) if n is even [7].Theorem. 1.1. The formally self adjoint fuchsian operator L 2.J2 = 250000 (6 ϑ + 5) (6 ϑ − 1) (3 ϑ − 1) (3 ϑ + 1) (6 ϑ + 1) (6 ϑ − 5) − 125 x (6 ϑ + 1) (6 ϑ + 5) · 1296000 ϑ 4 + 2592000 ϑ 3 + 2578320 ϑ 2 + 1282320 ϑ + 213703 + 11664 x 2 (10 ϑ + 17) (5 ϑ + 7) (10 ϑ + 11) (10 ϑ + 9) (5 ϑ + 3) (10 ϑ + 3) has the Riemann scheme and 2.J 2 as monodromy group. Moreover, its monodromy representation ρ : π 1 (P 1 \ {0, 1, ∞}, x 0 ) → Sp 6 (C) is uniquely determined by the local monodromy, i.e. the Jordan forms.2010 Mathematics Subject Classification. 34M50, 20C34, 32S40.