Permutation group homomorphism – sympy/numpy versio
Former-commit-id: 9fbc18dc9d2e4f22a5450e8e2aca799da45f5698
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Marek Nečada
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9e30722756
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#LyX 2.1 created this file. For more info see http://www.lyx.org/
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\pdf_title "Sähköpajan päiväkirja"
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\pdf_author "Marek Nečada"
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\begin_layout Title
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Symmetry-adapted basis functions for honeycomb lattice at
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\begin_inset Formula $K$
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\end_inset
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-point
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\begin_layout Section
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Generation theorem
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\begin_layout Standard
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Let
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be a group and
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\begin_inset Formula $\Gamma^{i}\left\{ R\to\mathbf{D}^{i}\left(R\right)\right\} $
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some
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\begin_inset Formula $d_{i}$
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-dimensional rep of
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\begin_inset Formula $\mathbf{G}$
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.
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Let the group ring (corresponding to the given rep indexed by
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) elements be defined as [Bradley&Cracknell (2.2.2)]
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\[
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W_{ts}^{i}=\frac{d_{i}}{\left|\mathbf{G}\right|}\sum_{R\in\mathbf{G}}\mathbf{D}^{i}\left(R\right)_{ts}^{*}R.
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\]
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From [Bradley&Cracknell, theorem 2.2.1]:
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If
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is an arbitrary function of
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(a linear space in which the realisation of the group operation act) such
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that
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(
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is fixed and is a number in the range 1 to
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;
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is idx of the rep) then the funs
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,
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to
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\begin_inset Formula $d_{i}$
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, form a basis for the rep
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\begin_inset Formula $\Gamma^{i}$
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.
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Particle-centered transformations
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Now let's see what are the point group actions on SVWF in the origin [Schulz]:
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Transformations in a lattice
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