The  matrices

Navaza, J.

doi:10.1107/97809553602060000682

International
Tables for
Crystallography
Volume F
Crystallography of biological macromolecules
Edited by M. G. Rossmann and E. Arnold

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International Tables for Crystallography (2006). Vol. F. ch. 13.2, pp. 273-274

Section A13.2.1.2. The $[D^{\ell}_{m, \, m'}]$ matrices

J. Navaza^a ^*

^aLaboratoire de Génétique des Virus, CNRS-GIF, 1. Avenue de la Terrasse, 91198 Gif-sur-Yvette, France
Correspondence e-mail: jnavaza@pasteur.fr

A13.2.1.2. The $[D^{\ell}_{m, \, m'}]$ matrices

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A linear representation of dimension n of the rotation group is a correspondence between rotations and matrices of order n. The matrices $[D^{\ell}_{m, \, m'}({\bf R})]$ , with $[-\ell \leq m,m' \leq \ell]$ , are associated with the irreducible representation of dimension $[2\ell + 1]$ ( $[0 \leq \ell \lt \infty]$ ). They have the following properties (Brink & Satchler, 1968):

(1) Group multiplication: $[D^{\ell}_{m, \, m'}({\bf RR'}) = {\textstyle\sum\limits_{n=-\ell}^{\ell}} D^{\ell}_{m, \, n}({\bf R}) D^{\ell}_{n, \, m'}({\bf R'}). \eqno\hbox{(A13.2.1.2)}]$
(2) Complex conjugation: $[D^{\ell}_{m, \, m'}({\bf R}^{-1}) = \overline{D^{\ell}_{m', \, m}({\bf R})}. \eqno\hbox{(A13.2.1.3)}]$
(3) Euler parameterization: $[D^{\ell}_{m, \, m'}(\alpha,\beta,\gamma) = d^{\ell}_{m, \, m'}(\beta) \exp [i(m\alpha+m'\gamma)]. \eqno\hbox{(A13.2.1.4)}]$
(4) Recurrence relation for the reduced matrices: $[\eqalignno{d^{\ell}_{m-1, \, m'}(\beta) &= {[m'-m \cos(\beta)]^{2} \over [(\ell-m+1)(\ell+m)]^{1/2} \sin(\beta)} d^{\ell}_{m, \, m'}(\beta)\cr &\quad - \left[{(\ell-m)(\ell+m+1) \over (\ell-m+1)(\ell+m)}\right]^{1/2} d^{\ell}_{m+1, \, m'}(\beta). &\hbox{(A13.2.1.5)}}]$
(5) Initial values (bottom row of $[d^{\ell}]$ ): $[d^{\ell}_{\ell, \, m}(\beta) = (-1)^{\ell-m} \left[{(2\ell)! \over (\ell-m)!(\ell+m)!}\right]^{1/2} \sin(\beta/2)^{\ell-m} \cos(\beta/2)^{\ell+m}. \eqno\hbox{(A13.2.1.6)}]$
(6) Symmetry relations: $[d^{\ell}_{-m, \, -m'}(\beta) = d^{\ell}_{m', \, m}(\beta) = (-1)^{m-m'} d^{\ell}_{m, \, m'}(\beta). \eqno\hbox{(A13.2.1.7)}]$

References

Brink, D. M. & Satchler, G. R. (1968). Angular momentum, 2nd ed. Oxford University Press.Google Scholar

International Tables for Crystallography (2006). Vol. F. ch. 13.2, pp. 273-274

Section A13.2.1.2. The matrices

A13.2.1.2. The matrices

References

Section A13.2.1.2. The $[D^{\ell}_{m, \, m'}]$ matrices

A13.2.1.2. The $[D^{\ell}_{m, \, m'}]$ matrices