<- [[:chaosbook]] ====== Chapter: World in a mirror ====== (ChaosBook.org blog, chapter [[http://chaosbook.org/paper.shtml#discrete|World in a mirror]]) --- //[[predrag.cvitanovic@physics.gatech.edu|Predrag Cvitanovic]] 2009-02-12// ===== Discrete symmetry desymmetrization ===== ==== Quotienting the discrete translation pCf isotropy subgroup ==== From Halcrow et al. paper on pCf equilibria: <latex> \begin{equation} \label{subg4RR} R_{xz} = \{e, \sigma_x \tau_{xz}, \sigma_z \tau_{xz}, \sigma_{xz}\} = \{e,\sigma_{xz}\} \times \{e,\sigma_{z}\tau_{xz}\} \simeq S \,. \end{equation} </latex> The <latex>R_{xz}</latex> isotropy subgroup is particularly important, as the equilibria belong to this conjugacy class, as do most of the solutions reported here. The //NBC// isotropy subgroup of Schmiegel and our //S// are conjugate to <latex>R_{xz}</latex> under quarter-cell coordinate transformations. In keeping with previous literature, we often represent this conjugacy class with <latex>S = \{e, s_1, s_2, s_3\} = \{e, \sigma_z \tau_x, \sigma_x \tau_{xz}, \sigma_{xz} \tau_z\}</latex> rather than the simpler conjugate group <latex>R_{xz}</latex>. {{gtspring2009:gibson.png?24}} Re. methods of visualizing the state-space portraits with the 4th-order <latex>R_{xz}</latex> isotropy subgroup quotiented out: the double-angle trick from Lorenz will not suffice here, since we have mirror symmetry <latex>(x,y,z) \to (-x,y,z)</latex> as well as the rotation-about axis <latex>(x,y,z) \to (-x,y,-z)</latex>. The double-angle trick is suitable only for the latter. It would reduce the four quadrants to two, but unfortunately not in the way we would like: it would map <latex>\tau_{xz} EQ2 to EQ2</latex> and <latex>\tau_z EQ2 \to \tau_x EQ2</latex>, leaving us with distinct <latex>EQ2, \tau_x EQ2</latex>. And it's <latex>EQ2, \tau_x EQ2</latex> we are most interested in equating. -- // John F. Gibson 2009-03-19//
