Representations of the Klein Group Determined by Quadruples of Polynomials Associated with the Double Confluent Heun Equation

Abstract

The canonical representation of the Klein group K₄ = ℤ₂⊕ℤ₂ on the space ℂ* = ℂ {0} induces a representation of this group on the ring L = C[z, z⁻¹], z ∈ ℂ*, of Laurent polynomials and, as a consequence, a representation of the group K₄ on the automorphism group of the group G = GL(4,L) by means of the elementwise action. The semidirect product ĜG = GK₄ is considered together with a realization of the group Ĝ as a group of semilinear automorphisms of the free 4-dimensional L-module M⁴. A three-parameter family of representations R of K₄ in the group Ĝ and a three-parameter family of elements X ∈ M⁴ with polynomial coordinates of degrees 2(ℓ − 1), 2ℓ, 2(ℓ − 1), and 2ℓ, where ℓ is an arbitrary positive integer (one of the three parameters), are constructed. It is shown that, for any given family of parameters, the vector X is a fixed point of the corresponding representation R. An algorithm for calculating the polynomials that are the components of X was obtained in a previous paper of the authors, in which it was proved that these polynomials give explicit formulas for automorphisms of the solution space of the doubly confluent Heun equation.