<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ FreeGResolution</code>( <var class="Arg">P</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a non-free <span class="SimpleMath">\(\mathbb ZG\)</span>-resolution <span class="SimpleMath">\(P_\ast\)</span> and a positive integer <span class="SimpleMath">\(n\)</span>. It attempts to return <span class="SimpleMath">\(n\)</span> terms of a free <span class="SimpleMath">\(\mathbb ZG\)</span>-resolution of <span class="SimpleMath">\(\mathbb Z\)</span>. However, the stabilizer groups in the non-free resolution must be such that HAP can construct free resolutions with contracting homotopies for them.</p>
<p>The contracting homotopy on the resolution was implemented by Bui Anh Tuan.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionBieberbachGroup</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionBieberbachGroup</code>( <var class="Arg">G</var>, <var class="Arg">v</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a torsion free crystallographic group <span class="SimpleMath">\(G\)</span>, also known as a Bieberbach group, represented using <strong class="button">AffineCrystGroupOnRight</strong> as in the GAP package Cryst. It also optionally inputs a choice of vector <span class="SimpleMath">\(v\)</span> in the Euclidean space <span class="SimpleMath">\(\mathbb R^n\)</span> on which <span class="SimpleMath">\(G\)</span> acts freely. The function returns <span class="SimpleMath">\(n+1\)</span> terms of the free ZG-resolution of <span class="SimpleMath">\(\mathbb Z\)</span> arising as the cellular chain complex of the tessellation of <span class="SimpleMath">\(\mathbb R^n\)</span> by the Dirichlet-Voronoi fundamental domain determined by <span class="SimpleMath">\(v\)</span>. No contracting homotopy is returned with the resolution.</p>
<p>This function is part of the HAPcryst package written by Marc Roeder and thus requires the HAPcryst package to be loaded.</p>
<p>The function requires the use of Polymake software.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionCubicalCrystGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a crystallographic group <span class="SimpleMath">\(G\)</span> represented using <strong class="button">AffineCrystGroupOnRight</strong> as in the GAP package <span class="SimpleMath">\(Cryst\)</span> together with an integer <span class="SimpleMath">\(k \ge 1\)</span>. The function tries to find a cubical fundamental domain in the Euclidean space <span class="SimpleMath">\(\mathbb R^n\)</span> on which <span class="SimpleMath">\(G\)</span> acts. If it succeeds it uses this domain to return <span class="SimpleMath">\(k+1\)</span> terms of a free ZG-resolution of <span class="SimpleMath">\(\mathbb Z\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionFiniteGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite group <span class="SimpleMath">\(G\)</span> and an integer <span class="SimpleMath">\(k \ge 1\)</span>. It returns <span class="SimpleMath">\(k+1\)</span> terms of a free ZG-resolution of <span class="SimpleMath">\(\mathbb Z\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionNilpotentGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a nilpotent group <span class="SimpleMath">\(G\)</span> (which can be infinite) and an integer <span class="SimpleMath">\(k \ge 1\)</span>. It returns <span class="SimpleMath">\(k+1\)</span> terms of a free <span class="SimpleMath">\(\mathbb ZG\)</span>-resolution of <span class="SimpleMath">\(\mathbb Z\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionNormalSeries</code>( <var class="Arg">L</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a a list <span class="SimpleMath">\(L\)</span> consisting of a chain $<span class="SimpleMath">\(1=N_1 \le N_2 \le \cdots \le N_n =G\)</span> of normal subgroups of <span class="SimpleMath">\(G\)</span>, together with an integer <span class="SimpleMath">\(k \ge 1\)</span>. It returns <span class="SimpleMath">\(k+1\)</span> terms of a free ZG-resolution of <span class="SimpleMath">\(\mathbb Z\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionPrimePowerGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(G\)</span> and an integer <span class="SimpleMath">\(k \ge 1\)</span>. It returns <span class="SimpleMath">\(k+1\)</span> terms of a minimal free <span class="SimpleMath">\(\mathbb FG\)</span>-resolution of the field <span class="SimpleMath">\(\mathbb F\)</span> of <span class="SimpleMath">\(p\)</span> elements.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSmallGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSmallGroup</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a small group <span class="SimpleMath">\(G\)</span> and an integer <span class="SimpleMath">\(k \ge 1\)</span>. It returns <span class="SimpleMath">\(k+1\)</span> terms of a free ZG-resolution of <span class="SimpleMath">\(\mathbb Z\)</span>.</p>
<p>If <span class="SimpleMath">\(G\)</span> is a finitely presented group then up to degree <span class="SimpleMath">\(2\)</span> the resolution coincides with cellular chain complex of the universal cover of the <span class="SimpleMath">\(2\)</span> complex associated to the presentation of <span class="SimpleMath">\(G\)</span>. Thus the boundaries of the generators in degree <span class="SimpleMath">\(3\)</span> provide a generating set for the module of identities of the presentation.</p>
<p>This function was written by Irina Kholodna.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ResolutionSubgroup</code>( <var class="Arg">R</var>, <var class="Arg">H</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a free ZG-resolution of <span class="SimpleMath">\(\mathbb Z\)</span> and a finite index subgroup <span class="SimpleMath">\(H \le G\)</span>. It returns a free ZH-resolution of <span class="SimpleMath">\(\mathbb Z\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LeibnizComplex</code>( <var class="Arg">g</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a Leibniz algebra, or Lie algebra, <span class="SimpleMath">\(\mathfrak{g}\)</span> over a ring <span class="SimpleMath">\(\mathbb K\)</span> together with an integer <span class="SimpleMath">\(n\ge 0\)</span>. It returns the first <span class="SimpleMath">\(n\)</span> terms of the Leibniz chain complex over <span class="SimpleMath">\(\mathbb K\)</span>. The complex was implemented by Pablo Fernandez Ascariz.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomToIntegralModule</code>( <var class="Arg">R</var>, <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a free <span class="SimpleMath">\(\mathbb ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> in characteristic <span class="SimpleMath">\(0\)</span> and a group homomorphism <span class="SimpleMath">\(A\colon G \rightarrow {\rm GL}_n(\mathbb Z)\)</span>. The homomorphism <span class="SimpleMath">\(A\)</span> can be viewed as the <span class="SimpleMath">\(\mathbb ZG\)</span>-module with underlying abelian group <span class="SimpleMath">\(\mathbb Z^n\)</span> on which <span class="SimpleMath">\(G\)</span> acts via the homomorphism <span class="SimpleMath">\(A\)</span>. It returns the cochain complex <span class="SimpleMath">\(Hom_{\mathbb ZG}(R,A)\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegers</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegers</code>( <var class="Arg">F</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a free <span class="SimpleMath">\(\mathbb ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> of characteristic <span class="SimpleMath">\(0\)</span> and returns the chain complex <span class="SimpleMath">\(R \otimes_{\mathbb ZG} {\mathbb Z}\)</span>.</p>
<p>Inputs an equivariant chain map <span class="SimpleMath">\(F\colon R \rightarrow S\)</span> in characteristic <span class="SimpleMath">\(0\)</span> and returns the induced chain map <span class="SimpleMath">\(F\otimes_{\mathbb ZG}\mathbb Z \colon R \otimes_{\mathbb ZG} {\mathbb Z} \longrightarrow S \otimes_{\mathbb ZG} {\mathbb Z}\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegersModP</code>( <var class="Arg">C</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegersModP</code>( <var class="Arg">R</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ TensorWithIntegersModP</code>( <var class="Arg">F</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a chain complex <span class="SimpleMath">\(C\)</span> of characteristic <span class="SimpleMath">\(0\)</span> and a prime integer <span class="SimpleMath">\(p\)</span>. It returns the chain complex <span class="SimpleMath">\(C \otimes_{\mathbb Z} {\mathbb Z}_p\)</span> of characteristic <span class="SimpleMath">\(p\)</span>.</p>
<p>Inputs a free <span class="SimpleMath">\(\mathbb ZG\)</span>-resolution <span class="SimpleMath">\(R\)</span> of characteristic <span class="SimpleMath">\(0\)</span> and a prime integer <span class="SimpleMath">\(p\)</span>. It returns the chain complex <span class="SimpleMath">\(R \otimes_{\mathbb ZG} {\mathbb Z}_p\)</span> of characteristic <span class="SimpleMath">\(p\)</span>.</p>
<p>Inputs an equivariant chain map <span class="SimpleMath">\(F\colon R \rightarrow S\)</span> in characteristic <span class="SimpleMath">\(0\)</span> a prime integer <span class="SimpleMath">\(p\)</span>. It returns the induced chain map <span class="SimpleMath">\(F\otimes_{\mathbb ZG}\mathbb Z_p \colon R \otimes_{\mathbb ZG} {\mathbb Z}_p \longrightarrow S \otimes_{\mathbb ZG} {\mathbb Z}_p\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ AreIsomorphicGradedAlgebras</code>( <var class="Arg">A</var>, <var class="Arg">B</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs two freely presented graded algebras <span class="SimpleMath">\(A=\mathbb F[x_1, \ldots, x_m]/I\)</span> and <span class="SimpleMath">\(B=\mathbb F[y_1, \ldots, y_n]/J\)</span> and returns <strong class="button">true</strong> if they are isomorphic, and <strong class="button">false</strong> otherwise. This function was implemented by Paul Smith.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HAPDerivation</code>( <var class="Arg">R</var>, <var class="Arg">I</var>, <var class="Arg">L</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a polynomial ring <span class="SimpleMath">\(R=\mathbb F[x_1,\ldots,x_m]\)</span> over a field <span class="SimpleMath">\(\mathbb F\)</span> together with a list <span class="SimpleMath">\(I\)</span> of generators for an ideal in <span class="SimpleMath">\(R\)</span> and a list <span class="SimpleMath">\(L=[y_1,\ldots,y_m]\subset R\)</span>. It returns the derivation <span class="SimpleMath">\(d\colon E \rightarrow E\)</span> for <span class="SimpleMath">\(E=R/I\)</span> defined by <span class="SimpleMath">\(d(x_i)=y_i\)</span>. This function was written by Paul Smith. It uses the Singular commutative algebra package.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HilbertPoincareSeries</code>( <var class="Arg">E</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a presentation <span class="SimpleMath">\(E=\mathbb F[x_1,\ldots,x_m]/I\)</span> of a graded algebra and returns its Hilbert–Poincaré series. This function was written by Paul Smith and uses the Singular commutative algebra package. It is essentially a wrapper for Singular's Hilbert–Poincaré series.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ HomologyOfDerivation</code>( <var class="Arg">d</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a derivation <span class="SimpleMath">\(d\colon E \rightarrow E\)</span> on a quotient <span class="SimpleMath">\(E=R/I\)</span> of a polynomial ring <span class="SimpleMath">\(R=\mathbb F[x_1,\ldots,x_m]\)</span> over a field <span class="SimpleMath">\(\mathbb F\)</span>. It returns a list <span class="SimpleMath">\([S,J,h]\)</span> where <span class="SimpleMath">\(S\)</span> is a polynomial ring and <span class="SimpleMath">\(J\)</span> is a list of generators for an ideal in <span class="SimpleMath">\(S\)</span> such that there is an isomorphism <span class="SimpleMath">\(\alpha\colon S/J \rightarrow \ker d/{\rm im~} d\)</span>. This isomorphism lifts to the ring homomorphism <span class="SimpleMath">\(h\colon S \rightarrow \ker d\)</span>. This function was written by Paul Smith. It uses the Singular commutative algebra package.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ IntegralCohomologyGenerators</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs at least <span class="SimpleMath">\(n+1\)</span> terms of a free <span class="SimpleMath">\(\mathbb ZG\)</span>-resolution of <span class="SimpleMath">\(\mathbb Z\)</span> and the integer <span class="SimpleMath">\( n \ge 1\)</span>. It returns a minimal list of cohomology classes in <span class="SimpleMath">\(H^n(G,\mathbb Z)\)</span> which, together with all cup products of lower degree classes, generate the group <span class="SimpleMath">\(H^n(G,\mathbb Z)\)</span> . (Let <span class="SimpleMath">\(a_i\)</span> be the <span class="SimpleMath">\(i\)</span>-th canonical generator of the <span class="SimpleMath">\(d\)</span>-generator abelian group <span class="SimpleMath">\(H^n(G,Z)\)</span>. The cohomology class <span class="SimpleMath">\(n_1a_1 + ... +n_da_d\)</span> is represented by the integer vector <span class="SimpleMath">\(u=(n_1, ..., n_d)\)</span>. )</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LHSSpectralSequence</code>( <var class="Arg">G</var>, <var class="Arg">N</var>, <var class="Arg">r</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">\(2\)</span>-group <span class="SimpleMath">\(G\)</span>, and normal subgroup <span class="SimpleMath">\(N\)</span> and an integer <span class="SimpleMath">\(r\)</span>. It returns a list of length <span class="SimpleMath">\(r\)</span> whose <spanclass="SimpleMath">\(i\)</span>-th term is a presentation for the <span class="SimpleMath">\(i\)</span>-th page of the Lyndon-Hochschild-Serre spectral sequence. This function was written by Paul Smith. It uses the Singular commutative algebra package.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ LHSSpectralSequenceLastSheet</code>( <var class="Arg">G</var>, <var class="Arg">N</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">\(2\)</span>-group <span class="SimpleMath">\(G\)</span> and normal subgroup <span class="SimpleMath">\(N\)</span>. It returns presentation for the <span class="SimpleMath">\(E_\infty\)</span> page of the Lyndon-Hochschild-Serre spectral sequence. This function was written by Paul Smith. It uses the Singular commutative algebra package.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModPCohomologyGenerators</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModPCohomologyGenerators</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(G\)</span> and positive integer <span class="SimpleMath">\(n\)</span>, or else <span class="SimpleMath">\(n+1\)</span> terms of a minimal <span class="SimpleMath">\(\mathbb FG\)</span>-resolution <span class="SimpleMath">\(R\)</span> of the field <span class="SimpleMath">\(\mathbb F\)</span> of <span class="SimpleMath">\(p\)</span> elements. It returns a pair whose first entry is a minimal list of homogeneous generators for the cohomology ring <span class="SimpleMath">\(A=H^\ast(G,\mathbb F)\)</span> modulo all elements in degree greater than <span class="SimpleMath">\(n\)</span>. The second entry of the pair is a function <strong class="button">deg</strong> which, when applied to a minimal generator, yields its degree. WARNING: the following rule must be applied when multiplying generators <span class="SimpleMath">\(x_i\)</span> together. Only products of the form <span class="SimpleMath">\(x_1*(x_2*(x_3*(x_4*...)))\)</span> with <span class="SimpleMath">\(deg(x_i) \le deg(x_{i+1})\)</span> should be computed (since the <span class="SimpleMath">\(x_i\)</span> belong to a structure constant algebra with only a partially defined structure constants table).</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModPCohomologyRing</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModPCohomologyRing</code>( <var class="Arg">R</var>, <var class="Arg">level</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModPCohomologyRing</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ ModPCohomologyRing</code>( <var class="Arg">G</var>, <var class="Arg">n</var>, <var class="Arg">level</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs either a <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(G\)</span> and positive integer <span class="SimpleMath">\(n\)</span>, or else <span class="SimpleMath">\(n\)</span> terms of a minimal <span class="SimpleMath">\(\mathbb FG\)</span>-resolution <span class="SimpleMath">\(R\)</span> of the field <span class="SimpleMath">\(\mathbb F\)</span> of <span class="SimpleMath">\(p\)</span> elements. It returns the cohomology ring <span class="SimpleMath">\(A=H^\ast(G,\mathbb F)\)</span> modulo all elements in degree greater than <span class="SimpleMath">\(n\)</span>. The ring is returned as a structure constant algebra <span class="SimpleMath">\(A\)</span>. The ring <span class="SimpleMath">\(A\)</span> is graded. It has a component <strong class="button">A!.degree(x)</strong> which is a function returning the degree of each (homogeneous) element <span class="SimpleMath">\(x\)</span> in <strong class="button">GeneratorsOfAlgebra(A)</strong>. An optional input variable <span class="SimpleMath">\("level"\)</span> can be set to one of the strings <span class="SimpleMath">\("medium"\)</span> or <span class="SimpleMath">\("high"\)</span>. These settings determine parameters in the algorithm. The default setting is <span class="SimpleMath">\("medium"\)</span>. When <span class="SimpleMath">\("level"\)</span> is set to <span class="SimpleMath">\("high"\)</span> the ring <span class="SimpleMath">\(A\)</span> is returned with a component <strong class="button">A!.niceBasis</strong>. This component is a pair <span class="SimpleMath">\([Coeff,Bas]\)</span>. Here <span class="SimpleMath">\(Bas\)</span> is a list of integer lists; a "nice" basis for the vector space <span class="SimpleMath">\(A\)</span> can be constructed using the command <strong class="button">List(Bas,x->Product(List(x,i->Basis(A)[i]))</strong>. The coefficients of the canonical basis element <strong class="button">Basis(A)[i]</strong> are stored as <strong class="button">Coeff[i]</strong>. If the ring <span class="SimpleMath">\(A\)</span> is computed using the setting <span class="SimpleMath">\("level"="medium"\)</span> then the component <strong class="button">A!.niceBasis</strong> can be added to <span class="SimpleMath">\(A\)</span> using the command <strong class="button">A:=ModPCohomologyRing_part_2(A)</strong>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mod2CohomologyRingPresentation</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mod2CohomologyRingPresentation</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mod2CohomologyRingPresentation</code>( <var class="Arg">A</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ Mod2CohomologyRingPresentation</code>( <var class="Arg">R</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>When applied to a finite <span class="SimpleMath">\(2\)</span>-group <span class="SimpleMath">\(G\)</span> this function returns a presentation for the mod-<span class="SimpleMath">\(2\)</span> cohomology ring <span class="SimpleMath">\(H^\ast(G,\mathbb F)\)</span>. The Lyndon-Hochschild-Serre spectral sequence is used to prove that the presentation is complete. When the function is applied to a <span class="SimpleMath">\(2\)</span>-group G and positive integer <span class="SimpleMath">\(n\)</span> the function first constructs <span class="SimpleMath">\(n+1\)</span> terms of a free <span class="SimpleMath">\(\mathbb FG\)</span>-resolution <span class="SimpleMath">\(R\)</span>, then constructs the finite-dimensional graded algebra <span class="SimpleMath">\(A=H^{(\ast \le n)}(G,\mathbb F)\)</span>, and finally uses <span class="SimpleMath">\(A\)</span> to approximate a presentation for <span class="SimpleMath">\(H^*(G,\mathbb F)\)</span>. For "sufficiently large" <span class="SimpleMath">\(n\)</span> the approximation will be a correct presentation for <span class="SimpleMath">\(H^\ast(G,\mathbb F)\)</span>. Alternatively, the function can be applied directly to either the resolution <span class="SimpleMath">\(R\)</span> or graded algebra <span class="SimpleMath">\(A\)</span>. This function was written by Paul Smith. It uses the Singular commutative algebra package to handle the Lyndon-Hochschild-Serre spectral sequence.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupCohomology</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupCohomology</code>( <var class="Arg">G</var>, <var class="Arg">k</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">\(G\)</span> and integer <span class="SimpleMath">\(k \ge 0\)</span>. The group <span class="SimpleMath">\(G\)</span> should either be finite or else lie in one of a range of classes of infinite groups (such as nilpotent, crystallographic, Artin etc.). The function returns the list of abelian invariants of <span class="SimpleMath">\(H^k(G,\mathbb Z)\)</span>.</p>
<p>If a prime <span class="SimpleMath">\(p\)</span> is given as an optional third input variable then the function returns the list of abelian invariants of <span class="SimpleMath">\(H^k(G,\mathbb Z_p)\)</span>. In this case each abelian invariant will be equal to <span class="SimpleMath">\(p\)</span> and the length of the list will be the dimension of the vector space <span class="SimpleMath">\(H^k(G,\mathbb Z_p)\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupHomology</code>( <var class="Arg">G</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ GroupHomology</code>( <var class="Arg">G</var>, <var class="Arg">k</var>, <var class="Arg">p</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">\(G\)</span> and integer <span class="SimpleMath">\(k \ge 0\)</span>. The group <span class="SimpleMath">\(G\)</span> should either be finite or else lie in one of a range of classes of infinite groups (such as nilpotent, crystallographic, Artin etc.). The function returns the list of abelian invariants of <span class="SimpleMath">\(H_k(G,\mathbb Z)\)</span>.</p>
<p>If a prime <span class="SimpleMath">\(p\)</span> is given as an optional third input variable then the function returns the list of abelian invariants of <span class="SimpleMath">\(H_k(G,\mathbb Z_p)\)</span>. In this case each abelian invariant will be equal to <span class="SimpleMath">\(p\)</span> and the length of the list will be the dimension of the vector space <span class="SimpleMath">\(H_k(G,\mathbb Z_p)\)</span>.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PrimePartDerivedFunctor</code>( <var class="Arg">G</var>, <var class="Arg">R</var>, <var class="Arg">A</var>, <var class="Arg">k</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a group <span class="SimpleMath">\(G\)</span>, an integer <span class="SimpleMath">\(k \ge 0\)</span>, at least <span class="SimpleMath">\(k+1\)</span> terms of a free <span class="SimpleMath">\(\mathbb ZP\)</span>-resolution of <span class="SimpleMath">\(\mathbb Z\)</span> for <span class="SimpleMath">\(P\)</span> a Sylow <span class="SimpleMath">\(p\)</span>-subgroup of <span class="SimpleMath">\(G\)</span>. A function such as <strong class="button">A=TensorWithIntegers</strong> is also entered. The abelian invariants of the <span class="SimpleMath">\(p\)</span>-primary part <span class="SimpleMath">\(H_k(G,A)_{(p)}\)</span> of the homology with coefficients in <span class="SimpleMath">\(A\)</span> is returned.</p>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">L</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(G\)</span> and a positive integer <span class="SimpleMath">\(n\)</span>. It returns a quotient of polynomials <span class="SimpleMath">\(f(x)=P(x)/Q(x)\)</span> whose expansion has coefficient of <span class="SimpleMath">\(x^k\)</span> equal to the rank of the vector space <span class="SimpleMath">\(H_k(G,\mathbb F_p)\)</span> for all <span class="SimpleMath">\(k\)</span> in the range <span class="SimpleMath">\(1 \le k \le n\)</span>. (The second input variable can be omitted, in which case the function tries to choose a `reasonable' value for \(n\). For 2-groups the function PoincareSeriesLHS(G) can be used to produce an \(f(x)\) that is correct in all degrees.) In place of the group \(G\) the function can also input (at least \(n\) terms of) a minimal mod-\(p\) resolution \(R\) for \(G\). Alternatively, the first input variable can be a list \(L\) of integers. In this case the coefficient of \(x^k\) in \(f(x)\) is equal to the \((k+1)\)st term in the list.
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">G</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">G</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">R</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<div class="func"><table class="func" width="100%"><tr><td class="tdleft"><code class="func">‣ PoincareSeries</code>( <var class="Arg">L</var>, <var class="Arg">n</var> )</td><td class="tdright">( function )</td></tr></table></div>
<p>Inputs a finite <span class="SimpleMath">\(p\)</span>-group <span class="SimpleMath">\(G\)</span> and a positive integer <span class="SimpleMath">\(n\)</span>. It returns a quotient of polynomials <span class="SimpleMath">\(f(x)=P(x)/Q(x)\)</span> whose expansion has coefficient of <span class="SimpleMath">\(x^k\)</span> equal to the rank of the vector space <span class="SimpleMath">\(H_k(G,\mathbb F_p)\)</span> for all <span class="SimpleMath">\(k\)</span> in the range <span class="SimpleMath">\(1 \le k \le n\)</span>. (The second input variable can be omitted, in which case the function tries to choose a `reasonable' value for \(n\). For 2-groups the function PoincareSeriesLHS(G) can be used to produce an \(f(x)\) that is correct in all degrees.) In place of the group \(G\) the function can also input (at least \(n\) terms of) a minimal mod-\(p\) resolution \(R\) for \(G\). Alternatively, the first input variable can be a list \(L\) of integers. In this case the coefficient of \(x^k\) in \(f(x)\) is equal to the \((k+1)\)st term in the list.
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