skip to main content
10.1145/1073884.1073889acmconferencesArticle/Chapter ViewAbstractPublication PagesissacConference Proceedingsconference-collections
Article

Signature of symmetric rational matrices and the unitary dual of lie groups

Published: 24 July 2005 Publication History

Abstract

A key step in the computation of the unitary dual of a Lie group is the determination if certain rational symmetric matrices are positive semi-definite. The size of some of the computations dictates that high performance integer matrix computations be used. We explore the feasibility of this approach by developing three algorithms for integer symmetric matrix signature and studying their performance both asymptotically and experimentally on a particular matrix family constructed from the exceptional Weyl group E8. We conclude that the computation is doable, with a parallel implementation needed for the largest representations.

References

[1]
J Adams. Integral models of representations of weyl groups. http://atlas.math.umd.edu/weyl/integral.
[2]
D. Barbasch. Unitary spherical spectrum for split classical groups, preprint. http://www.math.cornell.edu~barbasch/nsph.ps.
[3]
H. Bronnimann, I. Z. Emiris, V.Y. Pan, and S. Pion. Sign determination in residue number systems. Theoret. Comput. Sci., 210:173197, 1999.
[4]
L. Chen, W. Eberly, E. Kaltofen, B. D. Saunders, W. J. Turner, and G. Villard. Efficient matrix preconditioners for black box linear algebra. Linear Algebra and Applications, 343-344:119--146, 2002.
[5]
C.Pernet and Z. Wan. LU based algorithms for the characteristic polynomial over a finite field. In Poster, ISSAC'03. ACM Press, 2003.
[6]
Jean-Guillaume Dumas, Pascal Giorgi, and Cléement Pernet. FFPACK: Finite field linear algebra package.
[7]
Wayne Eberly. Early termination over small fields. In Proc. ISSAC'03, pages 80--87. ACM Press, 2003.
[8]
I. Z. Emiris. A complete implementation for computing general dimensional convex hulls. Inter. J. Comput. Geom. Appl., 8:223--253, 1998.
[9]
F. R. Gantmacher. The Theory of Matrices. Chelsea, New York, NY, 1959.
[10]
G. H. Golub and C. F. Van Loan. Matrix Computations. Johns Hopkins University Press, Baltimore, Maryland, third edition, 1996.
[11]
James E. Humphreys. Reflection groups and Coxeter groups, volume 29 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1990.
[12]
E. Kaltofen and A. Lobo. On rank properties of Toeplitz matrices overnite fields. In ISSAC 96 Proc. 1996 Internat. Symp. Symbolic Algebraic Comput., pages 241--249.
[13]
E. Kaltofen and B. D. Saunders. On Wiedemann's method of solving sparse linear systems. In H. F. Mattson, T. Mora, and T. R. N. Rao, editors, Proc. AAECC-9, volume 539 of Lect. Notes Comput. Sci., pages 29--38, Heidelberg, Germany, 1991. Springer Verlag.
[14]
Erich Kaltofen. An output-sensitive variant of the baby steps/giant steps determinant algorithm. In Proc. ISSAC'02, pages 138--144. ACM Press, 2002.
[15]
B. David Saunders and Zhendong Wan. Smith normal form of dense integer matrices fast algorithms into practice. In Proc. ISSAC'04, pages 274--281. ACM Press, 2004.
[16]
John R. Stembridge. Explicit matrices for irreducible representations of Weyl groups. Represent. Theory (electronic), 8:267--289, 2004.
[17]
D. Wiedemann. Solving sparse linear equations overnite fields. IEEE Trans. Inf. Theory, it-32:54--62, 1986.

Cited By

View all
  • (2011)On the generation of positivstellensatz witnesses in degenerate casesProceedings of the Second international conference on Interactive theorem proving10.5555/2033939.2033960(249-264)Online publication date: 22-Aug-2011
  • (2007)Numerical techniques for computing the inertia of products of matrices of rational numbersProceedings of the 2007 international workshop on Symbolic-numeric computation10.1145/1277500.1277520(125-132)Online publication date: 25-Jul-2007

Index Terms

  1. Signature of symmetric rational matrices and the unitary dual of lie groups

      Recommendations

      Comments

      Information & Contributors

      Information

      Published In

      cover image ACM Conferences
      ISSAC '05: Proceedings of the 2005 international symposium on Symbolic and algebraic computation
      July 2005
      388 pages
      ISBN:1595930957
      DOI:10.1145/1073884
      Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

      Sponsors

      Publisher

      Association for Computing Machinery

      New York, NY, United States

      Publication History

      Published: 24 July 2005

      Permissions

      Request permissions for this article.

      Check for updates

      Author Tags

      1. lie group
      2. matrix signature
      3. symmetric matrix

      Qualifiers

      • Article

      Conference

      ISSAC05
      Sponsor:

      Acceptance Rates

      Overall Acceptance Rate 395 of 838 submissions, 47%

      Contributors

      Other Metrics

      Bibliometrics & Citations

      Bibliometrics

      Article Metrics

      • Downloads (Last 12 months)0
      • Downloads (Last 6 weeks)0
      Reflects downloads up to 13 Feb 2025

      Other Metrics

      Citations

      Cited By

      View all
      • (2011)On the generation of positivstellensatz witnesses in degenerate casesProceedings of the Second international conference on Interactive theorem proving10.5555/2033939.2033960(249-264)Online publication date: 22-Aug-2011
      • (2007)Numerical techniques for computing the inertia of products of matrices of rational numbersProceedings of the 2007 international workshop on Symbolic-numeric computation10.1145/1277500.1277520(125-132)Online publication date: 25-Jul-2007

      View Options

      Login options

      View options

      PDF

      View or Download as a PDF file.

      PDF

      eReader

      View online with eReader.

      eReader

      Figures

      Tables

      Media

      Share

      Share

      Share this Publication link

      Share on social media