Abstract

For particles with discrete geometrical symmetries, a group-theoretical method is presented for transforming the matrix quantities in the T-matrix description of the electromagnetic scattering problem from the reducible basis of vector spherical wave functions into a new basis in which all matrix quantities become block diagonal. The notorious ill-conditioning problems in the inversion of the Q matrix are thus considerably alleviated, and the matrix inversion becomes numerically more expedient. The method can be applied to any point group. For the specific example of the D6h group, it is demonstrated that computations in the new basis are faster by a factor of 3.6 as compared with computations that use the reducible basis. Most importantly, the method is capable of extending the range of size parameters for which convergent results can be obtained by 50%.

© 2005 Optical Society of America

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  1. F. M. Schulz, K. Stamnes, J. J. Stamnes, “Point-group symmetries in electromagnetic scattering,” J. Opt. Soc. Am. A 16, 853–865 (1999).
    [CrossRef]
  2. F. M. Kahnert, J. J. Stamnes, K. Stamnes, “Application of the extended boundary condition method to homogeneous particles with point-group symmetries,” Appl. Opt. 40, 3110–3123 (2001).
    [CrossRef]
  3. F. M. Kahnert, J. J. Stamnes, K. Stamnes, “Can simple particle shapes be used to model scalar optical properties of an ensemble of wavelength-sized particles with complex shapes?,” J. Opt. Soc. Am. A 19, 521–531 (2002).
    [CrossRef]
  4. F. M. Kahnert, J. J. Stamnes, K. Stamnes, “Using simple particle shapes to model the Stokes scattering matrix of ensembles of wavelength-sized particles with complex shapes: possibilities and limitations,” J. Quant. Spectrosc. Radiat. Transf. 74, 167–182 (2002).
    [CrossRef]
  5. F. M. Kahnert, “Reproducing the optical properties of fine desert dust aerosols using ensembles of simple model particles,” J. Quant. Spectrosc. Radiat. Transf. 85, 231–249 (2004).
    [CrossRef]
  6. M. Kahnert, A. Kylling, “Radiance and flux simulations for mineral dust aerosols: assessing the error due to using spherical or spheroidal model particles,” J. Geophys. Res. 109, D09203 doi:10.1029/2003JD004318; errata; doi:10.1029/2004JD005311 (2004).
  7. T. Nousiainen, M. Kahnert, B. Veihelmann, “Light scattering modeling of small feldspar aerosol particles using polyhedral prisms and spheroids,” J. Quant. Spectrosc. Radiat. Transf. (to be published).
  8. I. A. Zagorodnov, R. P. Tarasov, “Finite groups in numerical solution of electromagnetic scattering problems on non-spherical particles,” in Light Scattering by Nonspherical Particles: Halifax Contributions (Army Research Laboratory, Adelphi, Md., 2000), pp. 99–102.
  9. T. Rother, M. Kahnert, A. Doicu, J. Wauer, “Surface Green’s function of the Helmholtz equation in spherical coordinates,” in Progress in Electromagnetic Research (PIER), J. A. Kong, ed. (EMW, Cambridge, Mass., 2002), Vol. 38, pp. 47–95.
    [CrossRef]
  10. D. M. Bishop, Group Theory and Chemistry (Dover, Mineola, N.Y., 1993).
  11. S. Havemann, A. J. Baran, “Extension of T matrix to scattering of electromagnetic plane waves by non-axisymmetric dielectric particles: application to hexagonal ice cylinders,” J. Quant. Spectrosc. Radiat. Transf. 70, 139–158 (2001).
    [CrossRef]
  12. M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).
  13. F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 775–824 (2003).
    [CrossRef]
  14. P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
    [CrossRef]
  15. F. M. Schulz, K. Stamnes, J. J. Stamnes, “Scattering of electromagnetic waves by spheroidal particles: a novel approach exploiting the T matrix computed in spheroidal coordinates,” Appl. Opt. 37, 7875–7896 (1998).
    [CrossRef]
  16. T. A. Niemen, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
    [CrossRef]
  17. D. W. Mackowski, “Discrete dipole moment method for computing the T matrix for nonspherical particles,” J. Opt. Soc. Am. A 19, 881–893 (2002).
    [CrossRef]
  18. D. W. Mackowski, M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13, 2266–2278 (1996).
    [CrossRef]
  19. J. D. Dixon, “High speed computation of group characters,” Numer. Math. 10, 446–450 (1965).
    [CrossRef]
  20. J. J. Cannon, “Computers in group theory: a survey,” Commun. ACM 12, 3–11 (1969).
    [CrossRef]
  21. D. C. Harris, M. D. Bertolucci, Symmetry and Spectroscopy (Oxford U. Press, New York, 1978).
  22. M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
    [CrossRef]
  23. S. Havemann, A. J. Baran, “Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method,” in Electromagnetic and Light Scattering—Theory and Applications VII, T. Wriedt, ed. (Universität Bremen, Bremen, Germany, 2003), pp. 107–110.
  24. D. J. Wielaard, M. I. Mishchenko, A. Macke, B. E. Carlson, “Improved T-matrix computations for large, nonabsorbing and weakly absorbing nonspherical particles and comparison with geometrical-optics approximation,” Appl. Opt. 36, 4305–4313 (1997).
    [CrossRef] [PubMed]
  25. M. Hamermesh, Group Theory and Its Application to Physical Problems (Dover, New York, 1989).

2004 (2)

F. M. Kahnert, “Reproducing the optical properties of fine desert dust aerosols using ensembles of simple model particles,” J. Quant. Spectrosc. Radiat. Transf. 85, 231–249 (2004).
[CrossRef]

M. Kahnert, A. Kylling, “Radiance and flux simulations for mineral dust aerosols: assessing the error due to using spherical or spheroidal model particles,” J. Geophys. Res. 109, D09203 doi:10.1029/2003JD004318; errata; doi:10.1029/2004JD005311 (2004).

2003 (2)

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 775–824 (2003).
[CrossRef]

T. A. Niemen, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[CrossRef]

2002 (3)

F. M. Kahnert, J. J. Stamnes, K. Stamnes, “Using simple particle shapes to model the Stokes scattering matrix of ensembles of wavelength-sized particles with complex shapes: possibilities and limitations,” J. Quant. Spectrosc. Radiat. Transf. 74, 167–182 (2002).
[CrossRef]

F. M. Kahnert, J. J. Stamnes, K. Stamnes, “Can simple particle shapes be used to model scalar optical properties of an ensemble of wavelength-sized particles with complex shapes?,” J. Opt. Soc. Am. A 19, 521–531 (2002).
[CrossRef]

D. W. Mackowski, “Discrete dipole moment method for computing the T matrix for nonspherical particles,” J. Opt. Soc. Am. A 19, 881–893 (2002).
[CrossRef]

2001 (2)

F. M. Kahnert, J. J. Stamnes, K. Stamnes, “Application of the extended boundary condition method to homogeneous particles with point-group symmetries,” Appl. Opt. 40, 3110–3123 (2001).
[CrossRef]

S. Havemann, A. J. Baran, “Extension of T matrix to scattering of electromagnetic plane waves by non-axisymmetric dielectric particles: application to hexagonal ice cylinders,” J. Quant. Spectrosc. Radiat. Transf. 70, 139–158 (2001).
[CrossRef]

1999 (1)

1998 (1)

1997 (1)

1996 (1)

1994 (1)

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

1969 (1)

J. J. Cannon, “Computers in group theory: a survey,” Commun. ACM 12, 3–11 (1969).
[CrossRef]

1965 (2)

J. D. Dixon, “High speed computation of group characters,” Numer. Math. 10, 446–450 (1965).
[CrossRef]

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Baran, A. J.

S. Havemann, A. J. Baran, “Extension of T matrix to scattering of electromagnetic plane waves by non-axisymmetric dielectric particles: application to hexagonal ice cylinders,” J. Quant. Spectrosc. Radiat. Transf. 70, 139–158 (2001).
[CrossRef]

S. Havemann, A. J. Baran, “Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method,” in Electromagnetic and Light Scattering—Theory and Applications VII, T. Wriedt, ed. (Universität Bremen, Bremen, Germany, 2003), pp. 107–110.

Bertolucci, M. D.

D. C. Harris, M. D. Bertolucci, Symmetry and Spectroscopy (Oxford U. Press, New York, 1978).

Bishop, D. M.

D. M. Bishop, Group Theory and Chemistry (Dover, Mineola, N.Y., 1993).

Cannon, J. J.

J. J. Cannon, “Computers in group theory: a survey,” Commun. ACM 12, 3–11 (1969).
[CrossRef]

Carlson, B. E.

Dixon, J. D.

J. D. Dixon, “High speed computation of group characters,” Numer. Math. 10, 446–450 (1965).
[CrossRef]

Doicu, A.

T. Rother, M. Kahnert, A. Doicu, J. Wauer, “Surface Green’s function of the Helmholtz equation in spherical coordinates,” in Progress in Electromagnetic Research (PIER), J. A. Kong, ed. (EMW, Cambridge, Mass., 2002), Vol. 38, pp. 47–95.
[CrossRef]

Hamermesh, M.

M. Hamermesh, Group Theory and Its Application to Physical Problems (Dover, New York, 1989).

Harris, D. C.

D. C. Harris, M. D. Bertolucci, Symmetry and Spectroscopy (Oxford U. Press, New York, 1978).

Havemann, S.

S. Havemann, A. J. Baran, “Extension of T matrix to scattering of electromagnetic plane waves by non-axisymmetric dielectric particles: application to hexagonal ice cylinders,” J. Quant. Spectrosc. Radiat. Transf. 70, 139–158 (2001).
[CrossRef]

S. Havemann, A. J. Baran, “Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method,” in Electromagnetic and Light Scattering—Theory and Applications VII, T. Wriedt, ed. (Universität Bremen, Bremen, Germany, 2003), pp. 107–110.

Heckenberg, N. R.

T. A. Niemen, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[CrossRef]

Kahnert, F. M.

F. M. Kahnert, “Reproducing the optical properties of fine desert dust aerosols using ensembles of simple model particles,” J. Quant. Spectrosc. Radiat. Transf. 85, 231–249 (2004).
[CrossRef]

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 775–824 (2003).
[CrossRef]

F. M. Kahnert, J. J. Stamnes, K. Stamnes, “Using simple particle shapes to model the Stokes scattering matrix of ensembles of wavelength-sized particles with complex shapes: possibilities and limitations,” J. Quant. Spectrosc. Radiat. Transf. 74, 167–182 (2002).
[CrossRef]

F. M. Kahnert, J. J. Stamnes, K. Stamnes, “Can simple particle shapes be used to model scalar optical properties of an ensemble of wavelength-sized particles with complex shapes?,” J. Opt. Soc. Am. A 19, 521–531 (2002).
[CrossRef]

F. M. Kahnert, J. J. Stamnes, K. Stamnes, “Application of the extended boundary condition method to homogeneous particles with point-group symmetries,” Appl. Opt. 40, 3110–3123 (2001).
[CrossRef]

Kahnert, M.

M. Kahnert, A. Kylling, “Radiance and flux simulations for mineral dust aerosols: assessing the error due to using spherical or spheroidal model particles,” J. Geophys. Res. 109, D09203 doi:10.1029/2003JD004318; errata; doi:10.1029/2004JD005311 (2004).

T. Nousiainen, M. Kahnert, B. Veihelmann, “Light scattering modeling of small feldspar aerosol particles using polyhedral prisms and spheroids,” J. Quant. Spectrosc. Radiat. Transf. (to be published).

T. Rother, M. Kahnert, A. Doicu, J. Wauer, “Surface Green’s function of the Helmholtz equation in spherical coordinates,” in Progress in Electromagnetic Research (PIER), J. A. Kong, ed. (EMW, Cambridge, Mass., 2002), Vol. 38, pp. 47–95.
[CrossRef]

Kylling, A.

M. Kahnert, A. Kylling, “Radiance and flux simulations for mineral dust aerosols: assessing the error due to using spherical or spheroidal model particles,” J. Geophys. Res. 109, D09203 doi:10.1029/2003JD004318; errata; doi:10.1029/2004JD005311 (2004).

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

Macke, A.

Mackowski, D. W.

Mishchenko, M. I.

Niemen, T. A.

T. A. Niemen, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[CrossRef]

Nousiainen, T.

T. Nousiainen, M. Kahnert, B. Veihelmann, “Light scattering modeling of small feldspar aerosol particles using polyhedral prisms and spheroids,” J. Quant. Spectrosc. Radiat. Transf. (to be published).

Rother, T.

T. Rother, M. Kahnert, A. Doicu, J. Wauer, “Surface Green’s function of the Helmholtz equation in spherical coordinates,” in Progress in Electromagnetic Research (PIER), J. A. Kong, ed. (EMW, Cambridge, Mass., 2002), Vol. 38, pp. 47–95.
[CrossRef]

Rubinsztein-Dunlop, H.

T. A. Niemen, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[CrossRef]

Schulz, F. M.

Stamnes, J. J.

Stamnes, K.

Tarasov, R. P.

I. A. Zagorodnov, R. P. Tarasov, “Finite groups in numerical solution of electromagnetic scattering problems on non-spherical particles,” in Light Scattering by Nonspherical Particles: Halifax Contributions (Army Research Laboratory, Adelphi, Md., 2000), pp. 99–102.

Travis, L. D.

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

Veihelmann, B.

T. Nousiainen, M. Kahnert, B. Veihelmann, “Light scattering modeling of small feldspar aerosol particles using polyhedral prisms and spheroids,” J. Quant. Spectrosc. Radiat. Transf. (to be published).

Waterman, P. C.

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Wauer, J.

T. Rother, M. Kahnert, A. Doicu, J. Wauer, “Surface Green’s function of the Helmholtz equation in spherical coordinates,” in Progress in Electromagnetic Research (PIER), J. A. Kong, ed. (EMW, Cambridge, Mass., 2002), Vol. 38, pp. 47–95.
[CrossRef]

Wielaard, D. J.

Zagorodnov, I. A.

I. A. Zagorodnov, R. P. Tarasov, “Finite groups in numerical solution of electromagnetic scattering problems on non-spherical particles,” in Light Scattering by Nonspherical Particles: Halifax Contributions (Army Research Laboratory, Adelphi, Md., 2000), pp. 99–102.

Appl. Opt. (3)

Commun. ACM (1)

J. J. Cannon, “Computers in group theory: a survey,” Commun. ACM 12, 3–11 (1969).
[CrossRef]

J. Geophys. Res. (1)

M. Kahnert, A. Kylling, “Radiance and flux simulations for mineral dust aerosols: assessing the error due to using spherical or spheroidal model particles,” J. Geophys. Res. 109, D09203 doi:10.1029/2003JD004318; errata; doi:10.1029/2004JD005311 (2004).

J. Opt. Soc. Am. A (4)

J. Quant. Spectrosc. Radiat. Transf. (5)

F. M. Kahnert, J. J. Stamnes, K. Stamnes, “Using simple particle shapes to model the Stokes scattering matrix of ensembles of wavelength-sized particles with complex shapes: possibilities and limitations,” J. Quant. Spectrosc. Radiat. Transf. 74, 167–182 (2002).
[CrossRef]

F. M. Kahnert, “Reproducing the optical properties of fine desert dust aerosols using ensembles of simple model particles,” J. Quant. Spectrosc. Radiat. Transf. 85, 231–249 (2004).
[CrossRef]

S. Havemann, A. J. Baran, “Extension of T matrix to scattering of electromagnetic plane waves by non-axisymmetric dielectric particles: application to hexagonal ice cylinders,” J. Quant. Spectrosc. Radiat. Transf. 70, 139–158 (2001).
[CrossRef]

F. M. Kahnert, “Numerical methods in electromagnetic scattering theory,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 775–824 (2003).
[CrossRef]

T. A. Niemen, H. Rubinsztein-Dunlop, N. R. Heckenberg, “Calculation of the T-matrix: general considerations and application of the point-matching method,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1019–1029 (2003).
[CrossRef]

Numer. Math. (1)

J. D. Dixon, “High speed computation of group characters,” Numer. Math. 10, 446–450 (1965).
[CrossRef]

Opt. Commun. (1)

M. I. Mishchenko, L. D. Travis, “T-matrix computations of light scattering by large spheroidal particles,” Opt. Commun. 109, 16–21 (1994).
[CrossRef]

Proc. IEEE (1)

P. C. Waterman, “Matrix formulation of electromagnetic scattering,” Proc. IEEE 53, 805–812 (1965).
[CrossRef]

Other (8)

S. Havemann, A. J. Baran, “Calculation of the phase matrix elements of elongated hexagonal ice columns using the T-matrix method,” in Electromagnetic and Light Scattering—Theory and Applications VII, T. Wriedt, ed. (Universität Bremen, Bremen, Germany, 2003), pp. 107–110.

M. Hamermesh, Group Theory and Its Application to Physical Problems (Dover, New York, 1989).

D. C. Harris, M. D. Bertolucci, Symmetry and Spectroscopy (Oxford U. Press, New York, 1978).

M. I. Mishchenko, L. D. Travis, A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge U. Press, Cambridge, UK, 2002).

T. Nousiainen, M. Kahnert, B. Veihelmann, “Light scattering modeling of small feldspar aerosol particles using polyhedral prisms and spheroids,” J. Quant. Spectrosc. Radiat. Transf. (to be published).

I. A. Zagorodnov, R. P. Tarasov, “Finite groups in numerical solution of electromagnetic scattering problems on non-spherical particles,” in Light Scattering by Nonspherical Particles: Halifax Contributions (Army Research Laboratory, Adelphi, Md., 2000), pp. 99–102.

T. Rother, M. Kahnert, A. Doicu, J. Wauer, “Surface Green’s function of the Helmholtz equation in spherical coordinates,” in Progress in Electromagnetic Research (PIER), J. A. Kong, ed. (EMW, Cambridge, Mass., 2002), Vol. 38, pp. 47–95.
[CrossRef]

D. M. Bishop, Group Theory and Chemistry (Dover, Mineola, N.Y., 1993).

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Figures (4)

Fig. 1
Fig. 1

Symmetry elements corresponding to the symmetry operations C 6 , C 2 ( 0 ) , C 2 ( 1 ) , C 2 ( 2 ) , C 2 ( 0 ) , C 2 ( 1 ) , C 2 ( 2 ) , and I.

Fig. 2
Fig. 2

Symmetry elements corresponding to the symmetry operations C 6 , σ v ( 0 ) , σ d ( 0 ) , σ h , and I.

Fig. 3
Fig. 3

Particle with a rotation–reflection symmetry element. Note that in this simple example the rotation–reflection operation is identical with the inversion operation I.

Fig. 4
Fig. 4

Left panel: phase function for a particle with D 6 h symmetry (as described in the text) computed with a T-matrix code that uses reducible representations of finite groups (solid curve) and a new implementation that uses irreducible representations of finite groups (dashed curve). Right panel: phase function for a larger D 6 h particle (as described in the text) computed with the new method. The old method did not converge in this test case.

Equations (95)

Equations on this page are rendered with MathJax. Learn more.

E inc ( k 0 r ) = m = M cut M cut n = m m + N cut [ a n , m , 1 ( M cut , N cut ) M n , m ( 1 ) ( k 0 r ) + a n , m , 2 ( M cut , N cut ) N n , m ( 1 ) ( k 0 r ) ] ,
E sca ( k 0 r ) = m = M cut M cut n = m m + N cut [ p n , m , 1 ( M cut , N cut ) M n , m ( 3 ) ( k 0 r ) + p n , m , 2 ( M cut , N cut ) N n , m ( 3 ) ( k 0 r ) ] ,
E int ( k r ) = m = M cut M cut n = m m + N cut [ c n , m , 1 ( M cut , N cut ) M n , m ( 1 ) ( k r ) + c n , m , 2 ( M cut , N cut ) N n , m ( 1 ) ( k r ) ]
n ̂ + × [ E inc ( k 0 r ) + E sca ( k 0 r ) E int ( k r ) ] r Γ = 0 ,
a n , m , τ = m = M cut M cut n = m m + N cut τ = 1 2 Q n , m , τ , n , m , τ c n , m , τ ,
p n , m , τ = m = M cut M cut n = m m + N cut τ = 1 2 R g Q n , m , τ , n , m , τ c n , m , τ ,
p n , m , τ = m = M cut M cut n = m m + N cut τ = 1 2 T n , m , τ , n , m , τ a n , m , τ ,
a = Q c ,
p = R g Q c ,
p = T a ,
T = R g Q Q 1 .
Q n , m , τ , n , m , τ = δ m , m Q n , m , τ , n , m , τ .
p n , m , τ = m = M cut M cut n = m m + N cut τ = 1 2 [ Σ ] n , m , τ , n , m , τ p n , m , τ
p = Σ p .
Q = Σ Q Σ 1 .
Σ Q Q Σ = 0 ;
[ E ] n , m , τ , n , m , τ = δ n , n δ m , m δ τ , τ .
[ C n j ] n , m , τ , n , m , τ = δ n , n δ m , m δ τ , τ exp ( i 2 π j m N ) ,
j = 1 , , N 1 .
[ C 2 ( j ) ] n , m , τ , n , m , τ = ( 1 ) n δ n , n δ m , m δ τ , τ exp ( i 4 π j m N ) ,
j = 0 , , P , P = { N 2 1 : N even N 1 : N odd } ,
[ C 2 ( j ) ] n , m , τ , n , m , τ = ( 1 ) n δ n , n δ m , m δ τ , τ exp [ i 4 π ( j + 1 2 ) m N ] ,
j = 0 , , N 2 1 .
[ σ v ( j ) ] n , m , τ , n , m , τ = ( 1 ) m + τ δ n , n δ m , m δ τ , τ exp ( i 4 π j m N ) ,
j = 0 , , P , P = { N 2 1 : N even N 1 : N odd } ,
[ σ d ( j ) ] n , m , τ , n , m , τ = ( 1 ) m + τ δ n , n δ m , m δ τ , τ exp [ i 4 π ( j + 1 2 ) m N ] ,
j = 0 , , N 2 1 .
[ σ h ] n , m , τ , n , m , τ = ( 1 ) n + m + τ δ n , n δ m , m δ τ , τ .
[ I ] n , m , τ , n , m , τ = ( 1 ) n + τ δ n , n δ m , m δ τ , τ .
[ S N ( j ) ] n , m , τ , n , m , τ = ( 1 ) n + m + τ δ n , n δ m , m δ τ , τ exp ( i 2 π j m N ) ,
j = 1 , , N 1 ,
D ( red ) = μ = 1 r α μ D ( μ ) .
ψ j ( μ , l ) g ψ j ( μ , l ) = i = 1 n μ D i , j ( μ ) ( g ) ψ i ( μ , l ) ,
μ = 1 , , k , l = 1 , , α μ .
ϕ j g ϕ j = i = 1 n D i j red ( g ) ϕ i .
ϕ i = μ = 1 k l = 1 α μ j = 1 n μ a i , j ( μ , l ) ψ j ( μ , l ) , i = 1 , , n .
ψ i ( μ , l ) = j = 1 n b i , j ( μ , l ) ϕ j ,
μ = 1 , , k , l = 1 , , α μ , i = 1 , , n μ .
j = 1 n b i , j ( μ , l ) a j , p ( ν , m ) = δ i , p δ μ , ν δ l , m .
D i , j red ( g ) = μ = 1 k l = 1 α μ p , q = 1 n μ a j , p ( μ , l ) D q , p ( μ ) ( g ) b q , i ( μ , l ) .
P ̃ j , i ( μ ) = g G χ ( μ ) * ( g ) D j , i red ( g )
P ̃ j , i ( μ ) = g G r = 1 n μ D r , r ( μ ) * ( g ) D j , i red ( g ) = ν = 1 k l = 1 α ν p , q = 1 n ν r = 1 n μ α i , p ( ν , l ) b q , j ( ν , l ) g G D r , r ( μ ) * ( g ) D q , p ( ν ) ( g ) = M o n μ r = 1 n μ l = 1 α μ a i , r ( μ , l ) b r , j ( μ , l ) .
j = 1 n P ̃ j , i ( μ ) ϕ j = M o n μ r = 1 n μ l = 1 α μ a i , r ( μ , l ) ψ r ( μ , l ) T μ .
D ( g ) = P 1 D red ( g ) P ,
D ( g ) = [ D ( 1 ) ( g ) D ( 2 ) ( g ) D ( k ) ( g ) ] .
D ( g ) Q = Q D ( g ) g G .
Q = [ Q ( 1 , 1 ) Q ( 1 , k ) Q ( k , 1 ) Q ( k , k ) ] .
D ( μ ) ( g ) Q ( μ , ν ) = Q ( μ , ν ) D ( ν ) ( g )
g G , μ , ν = 1 , , k .
D irr ( g ) = [ D ( 1 ) ( g ) D ( 1 ) ( g ) D ( k ) ( g ) D ( k ) ( g ) ] = [ U ( 1 ) ( g ) U ( k ) ( g ) ] ,
U ( μ ) ( g ) = [ D ( μ ) ( g ) D ( μ ) ( g ) ] , μ = 1 , , k ,
D ( μ ) ( g ) = S ( μ ) U ( μ ) ( g ) S 1 ( μ ) .
D ( μ ) ( g ) = [ D ( μ ; 1 , 1 ) D ( μ ; 1 , α μ ) D ( μ ; α μ , 1 ) D ( μ ; α μ , α μ ) ] .
D i , j ( μ ; l 1 , l 2 ) ( g ) = p , q = 1 n μ l 3 = 1 α μ S i , p ( μ ; l 1 , l 3 ) D p , q ( μ ) ( g ) ( S 1 ) q , j ( μ ; l 3 , l 2 ) .
Q ( μ , ν ) = [ Q ( μ , ν ; 1 , 1 ) Q ( μ , ν ; 1 , α ν ) Q ( μ , ν ; α μ , 1 ) Q ( μ , ν ; α μ , α ν ) ] ,
r = 1 n μ l 2 = 1 α μ D i , r ( μ ; l 1 , l 2 ) ( g ) Q r , j ( μ , ν ; l 2 , l 3 ) = s = 1 n ν l 4 = 1 α ν Q i , s ( μ , ν ; l 1 , l 4 ) D s , j ( ν ; l 4 , l 3 ) ( g ) ,
μ , ν = 1 , , k ,
i = 1 , , n μ , j = 1 , , n ν ,
l 1 = 1 , , α μ , l 3 = 1 , , α ν .
Q i , j ( μ , ν ; l 1 , l 3 ) = δ μ , ν Q i , j ( μ , μ ; l 1 , l 3 ) ,
μ , ν = 1 , , k ,
i = 1 , , n μ , j = 1 , , n ν ,
l 1 = 1 , , α μ , l 3 = 1 , , α ν .
p = 1 n μ D i , p ( μ ) ( g ) Q p , j ( μ ; l , n ) = q = 1 n μ Q i , q ( μ ; l , n ) D q , j ( μ ) ( g ) ,
E G ; g E = E g = g g G ,
g G g 1 G ; g g 1 = g 1 g = E ,
g 1 ( g 2 g 3 ) = ( g 1 g 2 ) g 3 g 1 , g 2 , g 3 G .
g 1 g 2 G g 1 , g 2 G .
g 1 = h 1 g 2 h .
[ g ] = { g G g g } = { g G h G ; g = h 1 g h }
D = α 1 D ( 1 ) α 2 D ( 2 ) α r D ( r ) ,
μ = 1 r n μ 2 = M o .
k = r .
g G D i , l ( μ ) ( g ) D m , j ( ν ) ( g 1 ) = M o n μ δ μ , ν δ i , j δ l , m .
D ( red ) = μ = 1 r α μ D ( μ ) .
α μ = M o 1 i = 1 k M i χ red ( g i ) χ ( μ ) * ( g i ) .
g G χ ( g ) χ * ( g ) = M o .
classes : E , σ h .
classes : E , I .
classes : E , C N , C N 2 , , C N N 1 .
M o = 2 N , k = { N 2 + 3 : N even N + 3 2 : N odd } ;
classes : { E , 2 C N , 2 C N 2 , , 2 C N N 2 1 , C N N 2 , N 2 C 2 , N 2 C 2 : N even E , 2 C N , , 2 C N ( N 1 ) 2 , N C 2 : N odd } .
M o = 2 N , k = { N 2 + 3 : N even N + 3 2 : N odd } ;
classes : { E , 2 C N , 2 C N 2 , , 2 C N N 2 1 , C N N 2 , N 2 σ v , N 2 σ d : N even E , 2 C N , , 2 C N ( N 1 ) 2 , N σ v : N odd } .
classes : { E , σ h , I , C N , C N 2 , , C N N 1 , S N , S N ( 2 ) , , S N ( N 2 1 ) , S N ( N 2 + 1 ) , , S N ( N 1 ) : N even E , σ h , C N , C N 2 , , C N N 1 , S N , S N ( 2 ) , , S N ( N 1 ) : N odd } .
M o = 4 N , k = { N + 6 : N even N + 3 : N odd } ;
classes : { E , σ h , I , 2 C N , 2 C N 2 , , 2 C N N 2 1 , C N N 2 , N 2 C 2 , N 2 C 2 , N 2 σ v , N 2 σ d , 2 S N 2 S N ( 2 ) , , 2 S N ( N 2 1 ) : N even E , σ h , 2 C N , 2 C N 2 , , 2 C N ( N 1 ) 2 , N C 2 , N σ v , 2 S N , 2 S N ( 2 ) , , 2 S N ( N 1 ) 2 : N odd } .
classes : { E , 2 C N , 2 C N 2 , , 2 C N N 2 1 , C N N 2 , N C 2 , N σ d , 2 S 2 N , 2 S 2 N ( 3 ) , , 2 S 2 N ( N 1 ) : N even E , I , 2 C N , 2 C N 2 , , 2 C N ( N 1 ) 2 , N C 2 , N σ d , 2 S 2 N , 2 S 2 N ( 3 ) , , 2 S 2 N ( N 2 ) : N odd } .
classes : { E , C N 2 , C N 2 2 , , C N 2 N 2 1 , S N , S N ( 3 ) , , S N ( N 1 ) : ( N 2 ) even E , I , C N 2 , C N 2 2 , , C N 2 N 2 1 , S N , S N ( 3 ) , , S N ( N 2 2 ) , S N ( N 2 + 2 ) , , S N ( N 1 ) : ( N 2 ) odd } .
classes : E , 4 C 3 , 4 C 3 2 , 3 C 2 .
classes : E , 8 C 3 , 3 C 2 , 6 S 4 , 6 σ d .
classes : E , I , 4 C 3 , 4 C 3 2 , 3 C 2 , 4 S 6 , 4 S 6 ( 2 ) , 3 σ h .
classes : E , 8 C 3 , 3 C 2 , 6 C 4 , 6 C 2 .
classes : E , 8 C 3 , 3 C 2 , 6 C 4 , 6 C 2 , I , 8 S 6 , 3 σ h , 6 S 4 , 6 σ d .
classes : E , 12 C 5 , 12 C 5 2 , 20 C 3 , 15 C 2 , I , 12 S 10 , 12 S 10 ( 3 ) , 20 S 6 , 15 σ .

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