Skip to main content
SpringerLink
Log in
Book cover

Singularity Theory and its Applications pp 80–106Cite as

Equivariant bifurcations and morsifications for finite groups

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 1463))

Abstract

For a bifurcation germ F(x,λ):ℝn+1,0→ℝn,0 which is equivariant with respect to the action of a finite group G, there are permutation actions of G on various subsets of branches of F−1(0). These sets include the set of all branches as well as the set of branches where λ>0 or<0 or where sign(det(dXF))>0 or <0, We shall give formulas for the modular characters of these permutation representations (which are the regular characters restricted to the odd order elements of G). These formulas are in terms of the representations of G on certain finite dimensional algebras associated to F. We deduce sufficient conditions for the existence of submaximal orbits by comparing the permutation representations for maximal orbits with certain representations of G.

Partially supported by a grant from the National Science Foundation and a Fulbright Fellowship

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   44.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   59.95
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliography

  1. Aoki, K., Fukuda, T. and Nishimura T. On the number of branches of the zero locus of a map germ (ℝn,0) → (ℝn−1,0), Topology and Computer Sciences, S. Suzuki ed. Kinokuniya Co. Ltd. Tokyo (1987) 347–367

    Google Scholar 

  2. Aoki, K., Fukuda, T. and Nishimura T. An algebraic formula for the topological types of one parameter bifurcation diagrams, Arch. Rat’l. Mech. and Anal. 108 (1989) 247–266

    Article  MathSciNet  MATH  Google Scholar 

  3. Chossat, P. Solutions avec symétrie diédrale dans les problèmes de bifurcation invariants par symétrie sphérique, Comptes Rendus ser 1 297 (1983) 639–642

    MathSciNet  MATH  Google Scholar 

  4. Cicogna, G. Symmetry breakdown from bifurcations, Letters al Nuovo Cimento 31 (1981) 600–602

    Article  MathSciNet  Google Scholar 

  5. Damon, J. Topological triviality and versality for subgroups of A and K II: Sufficient conditions and applications, preprint

    Google Scholar 

  6. — Topological equivalence of bifurcation problems, Nonlinearity 1 (1988) 311–331

    Article  MathSciNet  MATH  Google Scholar 

  7. — On the number of branches for real and complex weighted homogeneous curve singularities, to appear Topology

    Google Scholar 

  8. — G-signature, G-degree, and the symmetries of branches of curve singularities, to appear Topology

    Google Scholar 

  9. Dixon, J and Puttawamaiah, B.M. Modular Representations of Finite Groups, Pure and Applied Math. Series Academic Press (1977) New York-London

    Google Scholar 

  10. Eisenbud, D. and Levine, H. An algebraic formula for the degree of a C ∞ -map germ, Annals of Math. 106 (1977) 19–44

    Article  MathSciNet  MATH  Google Scholar 

  11. Field,M. Symmetry breaking for the Weyl groups W(D k ), preprint

    Google Scholar 

  12. Field, M. and Richardson, R.W. Symmetry Breaking and the maximal isotropy subgroup conjecture for reflection groups, Arch. Rat’l. Mech. and Anal. 105 (1989) 61–94

    MathSciNet  MATH  Google Scholar 

  13. Golubitsky, M. The Benard Problem, symmetry and the lattice of isotropy subgroups, in Bifurcation Theory, Mechanics and Physics, C. P. Bruter et al. eds., D. Reidel, Dordrecht-Boston-Lancaster (1983) 225–257

    Chapter  Google Scholar 

  14. Gusein-Zade, S. M. An Equivariant Analogue of the index of a gradient Vector Field, Springer Lecture Notes 1216 (1986) 196–210

    Google Scholar 

  15. Lauterbach, R. An example of symmetry breaking with submaximal isotropy, Contemp. Math. 56 (1986) 217–222

    Article  MathSciNet  MATH  Google Scholar 

  16. Montaldi, J. and van Straten, D. One-forms on singular curves and the topology of real curves singularities, to appear Topology

    Google Scholar 

  17. Roberts, M. Equivariant Milnor numbers and invariant Morse approximations, J. London Math. Soc. (2) 31 (1985) 487–500

    Article  MathSciNet  MATH  Google Scholar 

  18. Serre, J.P. Linear Representations of Finite Groups, Graduate Texts in Math. Springer-Verlag (19) Heidelberg-New York

    Google Scholar 

  19. Vanderbauwhede, A. Local Bifurcations and Symmetry, Pitman Research Notes in Math. 75(1982), Pitman Publ. London

    Google Scholar 

  20. Wahl, J. The Jacobian algebra of a graded Gorenstein singularity, Duke Jour. Math. 55 (1987) 843–871

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of North Carolina of North Carolina, 27599, Chapel Hill, N.C., U.S.A.

    James Damon

Authors
  1. James Damon
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Mark Roberts Ian Stewart

Rights and permissions

Reprints and permissions

Copyright information

© 1991 Springer-Verlag

About this paper

Cite this paper

Damon, J. (1991). Equivariant bifurcations and morsifications for finite groups. In: Roberts, M., Stewart, I. (eds) Singularity Theory and its Applications. Lecture Notes in Mathematics, vol 1463. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0085427

Download citation

  • DOI: https://doi.org/10.1007/BFb0085427

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-53736-6

  • Online ISBN: 978-3-540-47047-2

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation