Multidimensional Gauss reduction theory for conjugacy classes of SL ( n , )

Oleg Karpenkov[1]

  • [1] TU Graz 24, Kopernikusgasse 8010, Graz, Austria

Journal de Théorie des Nombres de Bordeaux (2013)

  • Volume: 25, Issue: 1, page 99-109
  • ISSN: 1246-7405

Abstract

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In this paper we describe the set of conjugacy classes in the group SL ( n , ) . We expand geometric Gauss Reduction Theory that solves the problem for SL ( 2 , ) to the multidimensional case, where ς -reduced Hessenberg matrices play the role of reduced matrices. Further we find complete invariants of conjugacy classes in GL ( n , ) in terms of multidimensional Klein-Voronoi continued fractions.

How to cite

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Karpenkov, Oleg. "Multidimensional Gauss reduction theory for conjugacy classes of $\mathrm{SL}(n,\mathbb{Z})$." Journal de Théorie des Nombres de Bordeaux 25.1 (2013): 99-109. <http://eudml.org/doc/275736>.

@article{Karpenkov2013,
abstract = {In this paper we describe the set of conjugacy classes in the group $\{\mathord \{\rm SL\}\}(n,\mathbb\{Z\})$. We expand geometric Gauss Reduction Theory that solves the problem for $\{\mathord \{\rm SL\}\}(2,\mathbb\{Z\})$ to the multidimensional case, where $\varsigma $-reduced Hessenberg matrices play the role of reduced matrices. Further we find complete invariants of conjugacy classes in $\{\mathord \{\rm GL\}\}(n,\mathbb\{Z\})$ in terms of multidimensional Klein-Voronoi continued fractions.},
affiliation = {TU Graz 24, Kopernikusgasse 8010, Graz, Austria},
author = {Karpenkov, Oleg},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {special linear group; Gauss reduction theory; multidimensional continued fraction},
language = {eng},
month = {4},
number = {1},
pages = {99-109},
publisher = {Société Arithmétique de Bordeaux},
title = {Multidimensional Gauss reduction theory for conjugacy classes of $\mathrm\{SL\}(n,\mathbb\{Z\})$},
url = {http://eudml.org/doc/275736},
volume = {25},
year = {2013},
}

TY - JOUR
AU - Karpenkov, Oleg
TI - Multidimensional Gauss reduction theory for conjugacy classes of $\mathrm{SL}(n,\mathbb{Z})$
JO - Journal de Théorie des Nombres de Bordeaux
DA - 2013/4//
PB - Société Arithmétique de Bordeaux
VL - 25
IS - 1
SP - 99
EP - 109
AB - In this paper we describe the set of conjugacy classes in the group ${\mathord {\rm SL}}(n,\mathbb{Z})$. We expand geometric Gauss Reduction Theory that solves the problem for ${\mathord {\rm SL}}(2,\mathbb{Z})$ to the multidimensional case, where $\varsigma $-reduced Hessenberg matrices play the role of reduced matrices. Further we find complete invariants of conjugacy classes in ${\mathord {\rm GL}}(n,\mathbb{Z})$ in terms of multidimensional Klein-Voronoi continued fractions.
LA - eng
KW - special linear group; Gauss reduction theory; multidimensional continued fraction
UR - http://eudml.org/doc/275736
ER -

References

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  9. F. Klein, Über eine geometrische Auffassung der gewöhnliche Kettenbruchentwicklung. Nachr. Ges. Wiss. Göttingen, Math-phys. Kl., 3:352–357, 1895. 
  10. E. I. Korkina, The simplest 2 -dimensional continued fraction. Topology, 3. J. Math. Sci. 82(5) (1996), 3680–3685. Zbl0901.11003MR1428725
  11. G. Lachaud, Voiles et polyhedres de Klein. Act. Sci. Ind., Hermann, 2002. 
  12. Y. I. Manin and M. Marcolli, Continued fractions, modular symbols, and noncommutative geometry. Selecta Math. (N.S.), 8(3) (2002), 475–521. Zbl1116.11033MR1931172
  13. G. F. Voronoĭ, Algorithm of the generalized continued fraction (In Russian). In Collected works in three volumes, volume I. USSR Ac. Sci., Kiev, 1952. 

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