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

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

top
In this paper we describe the set of conjugacy classes in the group SL ( n , ℤ ) . We expand geometric Gauss Reduction Theory that solves the problem for SL ( 2 , ℤ ) 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 , ℤ ) in terms of multidimensional Klein-Voronoi continued fractions.

How to cite

top

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

top
  1. H. Appelgate and H. Onishi, The similarity problem for 3 × 3 integer matrices. Linear Algebra Appl. 42 (1982), 159–174. Zbl0484.15011MR656422
  2. V. I. Arnold, Continued fractions (In Russian). Moscow Center of Continuous Mathematical Education, Moscow, 2002. 
  3. J. Buchmann, A generalization of Voronoĭ’s unit algorithm. I. J. Number Theory 20(2) (1985), 177–191. Zbl0575.12005MR790781
  4. J. Buchmann, A generalization of Voronoĭ’s unit algorithm. II. J. Number Theory 20(2) (1985), 192–209. Zbl0575.12005MR790782
  5. F. J. Grunewald, Solution of the conjugacy problem in certain arithmetic groups. Word problems, II (Conf. on Decision Problems in Algebra, Oxford, 1976), 95 of Stud. Logic Foundations Math., North-Holland, Amsterdam, 1980, 101–139. Zbl0447.20031MR579942
  6. O. Karpenkov, On the triangulations of tori associated with two-dimensional continued fractions of cubic irrationalities. Funct. Anal. Appl. 38(2) (2004), 102–110. Russian version: Funkt. Anal. Prilozh. 38(2) (2004), 28–37. Zbl1125.11042MR2086625
  7. O. Karpenkov, On Asymptotic Reducibility in S L ( 3 , ℤ ) . Technical report, arXiv:1205.4166, (2012). 
  8. S. Katok, Continued fractions, hyperbolic geometry and quadratic forms. In MASS selecta, Amer. Math. Soc., Providence, RI, 2003, 121–160. Zbl1091.11002MR2027174
  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. 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.