Integer conjugacy classes of SL(3,Z) and Hessenberg matrices

Details Integer conjugacy classes of SL(3,Z) and Hessenberg matrices Integer conjugacy classes of SL(3,Z) and Hessenberg matrices

2007-11-06

arxiv.org/abs/0711.0830v2

Mathematics

Number Theory

36 pages, 8 figures

Scientific paper

In this paper we study the problem of description of conjugacy classes in the group SL(n,Z). We expand Gauss Reduction Theory that gives the answer for the case n=2 to the multidimensional case. Reduced Hessenberg matrices now play the role of reduced matrices. For the case of three-dimensional matrices having a real and two complex-conjugate eigenvalues we show that perfect Hessenberg matrices distinguish conjugacy classes asymptotically. An important tool used in our approach is to determine minima of Markoff-Davenport characteristics at the vertices of Klein-Voronoi continued fractions. We conclude the paper with several open questions arising here.

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