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   KVecs: Γ, X, L, W</span></span></code></pre><p>which returns a <code>BandRepSet</code>, which itself is an <code>AbstractVector</code> of <code>BandRep</code>s. This allows us to index into <code>brs</code> easily:</p><pre><code class="language-julia hljs">brs[1] # obtain the EBR induced by Wyckoff position 8a with irrep A</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">2-band BandRep (A↑G at 8a):
  [Γ₁+Γ₂, X₁+X₂, L₁L₂, W₁W₂]</code></pre><p>By default, <code>bandreps</code> returns the spinless EBRs with time-reversal symmetry. This behavior can be controlled with the keyword arguments <code>spinful</code> (default, <code>false</code>) and <code>timereversal</code> (default, <code>true</code>). By default, only minimal paths are included in the sampling of <strong>k</strong>-vectors; additional paths can be obtained by setting the keyword argument <code>allpaths = true</code> (default, <code>false</code>).</p><p>The distinct topological classes identifiable from symmetry can can be calculated via <a href="../api/#Crystalline.classification-Tuple{AbstractVector{&lt;:Integer}}"><code>classification</code></a>, which uses the Smith normal form&#39;s principle factors:</p><pre><code class="language-julia hljs">classification(brs)</code></pre><pre class="documenter-example-output"><code class="nohighlight hljs ansi">&quot;Z₁&quot;</code></pre><p>Which demonstrates that the symmetry indicator group of spinless particles with time-reversal symmetry in space group 219 is trivial.</p><h2 id="Topology-and-associated-bases"><a class="docs-heading-anchor" href="#Topology-and-associated-bases">Topology and associated bases</a><a id="Topology-and-associated-bases-1"></a><a class="docs-heading-anchor-permalink" href="#Topology-and-associated-bases" title="Permalink"></a></h2><p>The <a href="https://github.com/thchr/SymmetryBases.jl"><code>SymmetryBases.jl</code></a> package provides tools to analyze topology of symmetry vectors and compute associated Hilbert bases.</p><h2 id="API"><a class="docs-heading-anchor" href="#API">API</a><a id="API-1"></a><a class="docs-heading-anchor-permalink" href="#API" title="Permalink"></a></h2><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Crystalline.bandreps-bandreps" href="#Crystalline.bandreps-bandreps"><code>Crystalline.bandreps</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">bandreps(sgnum::Integer, D::Integer=3; 
-         allpaths::Bool=false, spinful::Bool=false, timereversal::Bool=true)</code></pre><p>Returns the elementary band representations (EBRs) as a <code>BandRepSet</code> for space group <code>sgnum</code> and dimension <code>D</code>.</p><p><strong>Keyword arguments</strong></p><ul><li><code>allpaths</code>: include a minimal sufficient set (<code>false</code>, default) or all (<code>true</code>)              <strong>k</strong>-vectors. </li><li><code>spinful</code>: single- (<code>false</code>, default) or double-valued (<code>true</code>) irreps, as appropriate for            spinless and spinful particles, respectively. Only available for <code>D=3</code>.</li><li><code>timereversal</code>: assume presence (<code>true</code>, default) or absence (<code>false</code>) of time-reversal                 symmetry.</li></ul><p><strong>References</strong></p><p>3D EBRs are obtained from the Bilbao Crystallographic Server&#39;s  <a href="http://www.cryst.ehu.es/cgi-bin/cryst/programs/bandrep.pl">BANDREP program</a>; please reference the original research papers noted there if used in published work.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/thchr/Crystalline.jl/blob/38d82d5bb6dda3f373a99591ab85381c21bd1179/src/bandrep.jl#LL94-L114">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Crystalline.classification-bandreps" href="#Crystalline.classification-bandreps"><code>Crystalline.classification</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">classification(brs_or_F::Union{BandRepSet, Smith}) --&gt; String</code></pre><p>Return the symmetry indicator group <span>$X^{\text{BS}}$</span> of an EBR basis <code>F_or_brs</code>, provided as a <code>BandRepSet</code> or <code>Smith</code> decomposition.</p><p>Technically, the calculation answers the question &quot;what direct product of  <span>$\mathbb{Z}_n$</span> groups is the the quotient group <span>$X^{\text{BS}} = \{\text{BS}\}/\{\text{AI}\}$</span> isomorphic to?&quot; (see <a href="https://doi.org/10.1038/s41467-017-00133-2">Po, Watanabe, &amp; Vishwanath, Nature Commun. <strong>8</strong>, 50 (2017)</a> for more information).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/thchr/Crystalline.jl/blob/38d82d5bb6dda3f373a99591ab85381c21bd1179/src/bandrep.jl#LL145-L156">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Crystalline.nontrivial_factors-bandreps" href="#Crystalline.nontrivial_factors-bandreps"><code>Crystalline.nontrivial_factors</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">nontrivial_factors(F::Smith) -&gt; Any
-</code></pre><p>Return the nontrivial (i.e., ≠ {0,1}) elementary factors of an EBR basis, provided as a <code>BandRepSet</code> or <code>Smith</code> decomposition.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/thchr/Crystalline.jl/blob/38d82d5bb6dda3f373a99591ab85381c21bd1179/src/bandrep.jl#LL128">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Crystalline.basisdim-bandreps" href="#Crystalline.basisdim-bandreps"><code>Crystalline.basisdim</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">basisdim(brs::BandRepSet) --&gt; Int</code></pre><p>Return the dimension of the (linearly independent parts) of a band representation set. This is <span>$d^{\text{bs}} = d^{\text{ai}}$</span> in the notation of <a href="https://doi.org/10.1038/s41467-017-00133-2">Po, Watanabe, &amp; Vishwanath, Nature Commun. <strong>8</strong>, 50 (2017)</a>, or  equivalently, the rank of <code>stack(brs)</code> over the ring of integers. This is the number of linearly independent basis vectors that span the expansions of a band structure viewed as symmetry data.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/thchr/Crystalline.jl/blob/38d82d5bb6dda3f373a99591ab85381c21bd1179/src/bandrep.jl#LL168-L177">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../bravais/">« Bravais types &amp; bases</a><a class="docs-footer-nextpage" href="../lattices/">Lattices »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Thursday 12 December 2024 14:09">Thursday 12 December 2024</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>
+         allpaths::Bool=false, spinful::Bool=false, timereversal::Bool=true)</code></pre><p>Returns the elementary band representations (EBRs) as a <code>BandRepSet</code> for space group <code>sgnum</code> and dimension <code>D</code>.</p><p><strong>Keyword arguments</strong></p><ul><li><code>allpaths</code>: include a minimal sufficient set (<code>false</code>, default) or all (<code>true</code>)              <strong>k</strong>-vectors. </li><li><code>spinful</code>: single- (<code>false</code>, default) or double-valued (<code>true</code>) irreps, as appropriate for            spinless and spinful particles, respectively. Only available for <code>D=3</code>.</li><li><code>timereversal</code>: assume presence (<code>true</code>, default) or absence (<code>false</code>) of time-reversal                 symmetry.</li></ul><p><strong>References</strong></p><p>3D EBRs are obtained from the Bilbao Crystallographic Server&#39;s  <a href="http://www.cryst.ehu.es/cgi-bin/cryst/programs/bandrep.pl">BANDREP program</a>; please reference the original research papers noted there if used in published work.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/thchr/Crystalline.jl/blob/7337a269deee9759752d6dc1abeab0f15820101c/src/bandrep.jl#LL94-L114">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Crystalline.classification-bandreps" href="#Crystalline.classification-bandreps"><code>Crystalline.classification</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">classification(brs_or_F::Union{BandRepSet, Smith}) --&gt; String</code></pre><p>Return the symmetry indicator group <span>$X^{\text{BS}}$</span> of an EBR basis <code>F_or_brs</code>, provided as a <code>BandRepSet</code> or <code>Smith</code> decomposition.</p><p>Technically, the calculation answers the question &quot;what direct product of  <span>$\mathbb{Z}_n$</span> groups is the the quotient group <span>$X^{\text{BS}} = \{\text{BS}\}/\{\text{AI}\}$</span> isomorphic to?&quot; (see <a href="https://doi.org/10.1038/s41467-017-00133-2">Po, Watanabe, &amp; Vishwanath, Nature Commun. <strong>8</strong>, 50 (2017)</a> for more information).</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/thchr/Crystalline.jl/blob/7337a269deee9759752d6dc1abeab0f15820101c/src/bandrep.jl#LL145-L156">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Crystalline.nontrivial_factors-bandreps" href="#Crystalline.nontrivial_factors-bandreps"><code>Crystalline.nontrivial_factors</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">nontrivial_factors(F::Smith) -&gt; Any
+</code></pre><p>Return the nontrivial (i.e., ≠ {0,1}) elementary factors of an EBR basis, provided as a <code>BandRepSet</code> or <code>Smith</code> decomposition.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/thchr/Crystalline.jl/blob/7337a269deee9759752d6dc1abeab0f15820101c/src/bandrep.jl#LL128">source</a></section></article><article class="docstring"><header><a class="docstring-article-toggle-button fa-solid fa-chevron-down" href="javascript:;" title="Collapse docstring"></a><a class="docstring-binding" id="Crystalline.basisdim-bandreps" href="#Crystalline.basisdim-bandreps"><code>Crystalline.basisdim</code></a> — <span class="docstring-category">Function</span><span class="is-flex-grow-1 docstring-article-toggle-button" title="Collapse docstring"></span></header><section><div><pre><code class="language-julia hljs">basisdim(brs::BandRepSet) --&gt; Int</code></pre><p>Return the dimension of the (linearly independent parts) of a band representation set. This is <span>$d^{\text{bs}} = d^{\text{ai}}$</span> in the notation of <a href="https://doi.org/10.1038/s41467-017-00133-2">Po, Watanabe, &amp; Vishwanath, Nature Commun. <strong>8</strong>, 50 (2017)</a>, or  equivalently, the rank of <code>stack(brs)</code> over the ring of integers. This is the number of linearly independent basis vectors that span the expansions of a band structure viewed as symmetry data.</p></div><a class="docs-sourcelink" target="_blank" href="https://github.com/thchr/Crystalline.jl/blob/7337a269deee9759752d6dc1abeab0f15820101c/src/bandrep.jl#LL168-L177">source</a></section></article></article><nav class="docs-footer"><a class="docs-footer-prevpage" href="../bravais/">« Bravais types &amp; bases</a><a class="docs-footer-nextpage" href="../lattices/">Lattices »</a><div class="flexbox-break"></div><p class="footer-message">Powered by <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> and the <a href="https://julialang.org/">Julia Programming Language</a>.</p></nav></div><div class="modal" id="documenter-settings"><div class="modal-background"></div><div class="modal-card"><header class="modal-card-head"><p class="modal-card-title">Settings</p><button class="delete"></button></header><section class="modal-card-body"><p><label class="label">Theme</label><div class="select"><select id="documenter-themepicker"><option value="auto">Automatic (OS)</option><option value="documenter-light">documenter-light</option><option value="documenter-dark">documenter-dark</option><option value="catppuccin-latte">catppuccin-latte</option><option value="catppuccin-frappe">catppuccin-frappe</option><option value="catppuccin-macchiato">catppuccin-macchiato</option><option value="catppuccin-mocha">catppuccin-mocha</option></select></div></p><hr/><p>This document was generated with <a href="https://github.com/JuliaDocs/Documenter.jl">Documenter.jl</a> version 1.8.0 on <span class="colophon-date" title="Monday 16 December 2024 09:12">Monday 16 December 2024</span>. Using Julia version 1.11.2.</p></section><footer class="modal-card-foot"></footer></div></div></div></body></html>