| Quellcodebibliothek Statistik Leitseite products/Sources/formale Sprachen/GAP/pkg/repndecomp/lib/ (GAP Algebra Version 4.15.1©) Datei vom 12.8.2025 mit Größe 1 kB |
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Quelle groupsum.gd Sprache: Shell |
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#! @Chapter Miscellaneous useful functions #! @Section Efficient summing over groups #! @Arguments G, summand #! @Returns $\sum_{g \in G} \mbox{summand}(g)$ #! @Description Uses a basic stabiliser chain for $G$ to compute the #! sum described above. This trick requires <A>summand</A> to be a #! function (in the GAP sense) that defines a monoid homomorphism (in #! the mathematical sense). The computation of the stabiliser chain #! assumes <A>G</A> is a group. #! More precisely, if we have the basic stabiliser chain: #! $$\{1\} = G_1 \leq \ldots \leq G_n = G$$ #! We traverse the chain from $G_1$ to $G_n$, using the previous sum #! $G_{i-1}$ to build the sum $G_i$. We do this by using the fact #! that (writing $f$ for <A>summand</A>) #! $$\sum_{g \in G_i} f(g) = \sum_{r_j} \left(\sum_{h \in G_{i-1}} f(h)\right) f(r_j)$$ #! where the $r_j$ are right coset representatives of $G_{i-1}$ in #! $G_i$. #! The condition on <A>summand</A> is satisfied if, for example, it is #! a linear representation of a group <A>G</A>. DeclareGlobalFunction( "GroupSumBSGS" );
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2026-11-01