Permutation group homomorphism – sympy/numpy versio

Former-commit-id: 9fbc18dc9d2e4f22a5450e8e2aca799da45f5698

notes/hexlattice_kpoint_projections.lyx

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+\use_hyperref true
+\pdf_title "Sähköpajan päiväkirja"
+\pdf_author "Marek Nečada"
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+\index Index
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+
+\begin_body
+
+\begin_layout Title
+Symmetry-adapted basis functions for honeycomb lattice at 
+\begin_inset Formula $K$
+\end_inset
+
+-point
+\end_layout
+
+\begin_layout Section
+Generation theorem
+\end_layout
+
+\begin_layout Standard
+Let 
+\begin_inset Formula $\mathbf{G}$
+\end_inset
+
+ be a group and 
+\begin_inset Formula $\Gamma^{i}\left\{ R\to\mathbf{D}^{i}\left(R\right)\right\} $
+\end_inset
+
+ some 
+\begin_inset Formula $d_{i}$
+\end_inset
+
+-dimensional rep of 
+\begin_inset Formula $\mathbf{G}$
+\end_inset
+
+.
+ Let the group ring (corresponding to the given rep indexed by 
+\begin_inset Formula $i$
+\end_inset
+
+) elements be defined as [Bradley&Cracknell (2.2.2)]
+\begin_inset Formula 
+\[
+W_{ts}^{i}=\frac{d_{i}}{\left|\mathbf{G}\right|}\sum_{R\in\mathbf{G}}\mathbf{D}^{i}\left(R\right)_{ts}^{*}R.
+\]
+
+\end_inset
+
+
+\end_layout
+
+\begin_layout Standard
+From [Bradley&Cracknell, theorem 2.2.1]:
+\end_layout
+
+\begin_layout Standard
+If 
+\begin_inset Formula $\phi$
+\end_inset
+
+ is an arbitrary function of 
+\begin_inset Formula $V$
+\end_inset
+
+ (a linear space in which the realisation of the group operation act) such
+ that 
+\begin_inset Formula $W_{ss}^{i}\phi\ne0$
+\end_inset
+
+ (
+\begin_inset Formula $s$
+\end_inset
+
+ is fixed and is a number in the range 1 to 
+\begin_inset Formula $d_{i}$
+\end_inset
+
+; 
+\begin_inset Formula $i$
+\end_inset
+
+ is idx of the rep) then the funs 
+\begin_inset Formula $W_{ts}^{i}\phi=\phi_{t}^{i}$
+\end_inset
+
+, 
+\begin_inset Formula $t=1$
+\end_inset
+
+ to 
+\begin_inset Formula $d_{i}$
+\end_inset
+
+, form a basis for the rep 
+\begin_inset Formula $\Gamma^{i}$
+\end_inset
+
+.
+\end_layout
+
+\begin_layout Section
+Particle-centered transformations
+\end_layout
+
+\begin_layout Standard
+Now let's see what are the point group actions on SVWF in the origin [Schulz]:
+\end_layout
+
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+\begin_layout Section
+Transformations in a lattice
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+
+\begin_layout Standard
+
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+
+\end_body
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