Illinois Journal of Mathematics

Fixed points in absolutely irreducible real representations

Haibo Ruan

Full-text: Open access

Abstract

It has been an open question whether any bifurcation problem with absolutely irreducible group action would lead to bifurcation of steady states. A positive proposal is also known as the “Ize-conjecture”. Algebraically speaking, this is to ask whether every absolutely irreducible real representation has an odd dimensional fixed point subspace corresponding to some subgroups. Recently, Reiner Lauterbach and Paul Matthews have found counter examples to this conjecture and interestingly, all of the representations are of dimension $4k$, for $k\in\mathbb{N}$. A natural question arises: what about the case $4k+2$?

In this paper, we give a partial answer to this question and prove that in any $6$-dimensional absolutely irreducible real representation of a finite solvable group, there exists an odd dimensional fixed point subspace with respect to subgroups.

Article information

Source
Illinois J. Math., Volume 55, Number 4 (2011), 1551-1567.

Dates
First available in Project Euclid: 12 July 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1373636696

Digital Object Identifier
doi:10.1215/ijm/1373636696

Mathematical Reviews number (MathSciNet)
MR3082881

Zentralblatt MATH identifier
1279.20018

Subjects
Primary: 20C30: Representations of finite symmetric groups 19A22: Frobenius induction, Burnside and representation rings
Secondary: 37G40: Symmetries, equivariant bifurcation theory 37C25: Fixed points, periodic points, fixed-point index theory

Citation

Ruan, Haibo. Fixed points in absolutely irreducible real representations. Illinois J. Math. 55 (2011), no. 4, 1551--1567. doi:10.1215/ijm/1373636696. https://projecteuclid.org/euclid.ijm/1373636696


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References

  • Z. Balanov, W. Krawcewicz and H. Steinlein, Applied equivariant degree, AIMS Series on Differential Equations & Dynamical Systems, vol. 1, American Institute of Mathematical Sciences (AIMS), Springfield, 2006.
  • T. Bröcker and T. tom Dieck, Representations of compact Lie groups, Graduate Texts in Mathematics, vol. 98, Springer-Verlag, New York, 1985.
  • T. tom Dieck, Transformation groups, de Gruyter, Berlin, 1987.
  • M. Field, Symmetry breaking for compact Lie groups, Mem. Amer. Math. Soc., vol. 120, Amer. Math. Soc., Providence, 1996.
  • I. M. Isaacs, Character theory of finite groups, Academic Press, New York, 1976.
  • I. M. Isaacs, Characters of groups having fixed-point-free automorphisms of 2-power order, J. Algebra 328 (2011), no. 1, 218–229.
  • J. Ize, Private communication.
  • K. Kawakubo, The theory of transformation groups, Oxford University Press, New York, 1991.
  • W. Krawcewicz and J. Wu, Theory of degrees with applications to bifurcations and differential equations, Canadian Mathematical Society Series of Monographs and Advanced Texts, Wiley, New York, 1997.
  • A. Kushkuley and Z. Balanov, Geometric methods in degree theory for equivariant maps, Lecture Notes in Math., vol. 1632, Springer-Verlag, Berlin, 1996.
  • R. Lauterbach and P. Matthews, Do absolutely irreducible group actions have odd dimensional fixed point spaces?, unpublished manuscript, available at http://arxiv.org/abs/1011.3986.
  • H. Ulrich, Fixed point theory of parametrized equivariant maps, Lecture Notes in Math., vol. 1343, Springer-Verlag, Berlin, 1988.