Thetafunktionen und konjugationsinvariante Funktionen auf Paaren von Matrizen

Abstract

We express conjugation-invariant functions on the space of equivalence classes of pairs of kxk matrices via theta functions. Our approach is based on the well-known interplay between algebraic geometry and the theory of integrable systems. Every equivalence class of pairs of matrices defines the so-called spectral curve S and a line bundle L on S. Both together give a complete invariant for equivalence classes of pairs of matrices. This yields an identification between the set of equivalence classes of pairs of matrices with fixed spectral curve and the affine Jacobian of this curve, i.e. the Jacobian without a theta divisor. By using Painleve-analysis we describe subvarieties of the theta divisor by families of formal Laurent series of matrix polynomials. This leads to a description of conjugation invariant functions by theta functions modulo constant terms. To determine the explicit constant term we develop concrete extensions of sections of appropriate vector bundles of arbitrary order. Finally we present an algorithm for describing conjugation invariant functions on pairs of matrices via theta functions and carry out the calculations on the example tr([A,B]2).