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tutorial:apps_group [2017/08/07 11:30] – [Quotient spaces] commented out expensive homology computation to reduce testing time oroehrig | user_guide:tutorials:apps_group [2019/01/25 13:40] – ↷ Links adapted because of a move operation oroehrig |
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We start with the description of permutation groups in ''polymake''. An object of type [[https://polymake.org/release_docs/latest/group.html#group__Group__5 | Group]] records the abstract properties of the groups that do not depend on any particular representation, which essentially are just the ''ORDER'', ''CHARACTER_TABLE'', and ''CONJUGACY_CLASS_SIZES''. Moreover, a ''Group'' object can contain several subobjects that encode actions (representations) of the group, most notably a ''PERMUTATION_ACTION'' that encodes permutations of indices. If the Group object is contained inside a ''Cone'', ''Polytope'', ''PointConfiguration'', or ''VectorConfiguration'', it may be encoded more specifically as a ''RAY_ACTION'', ''FACET_ACTION'', etc. See the [[https://polymake.org/doku.php/release_docs/latest/group.html | documentation]] for more information on the action types. | We start with the description of permutation groups in ''polymake''. An object of type [[https://polymake.org/release_docs/latest/group.html#group__Group__5 | Group]] records the abstract properties of the groups that do not depend on any particular representation, which essentially are just the ''ORDER'', ''CHARACTER_TABLE'', and ''CONJUGACY_CLASS_SIZES''. Moreover, a ''Group'' object can contain several subobjects that encode actions (representations) of the group, most notably a ''PERMUTATION_ACTION'' that encodes permutations of indices. If the Group object is contained inside a ''Cone'', ''Polytope'', ''PointConfiguration'', or ''VectorConfiguration'', it may be encoded more specifically as a ''RAY_ACTION'', ''FACET_ACTION'', etc. See the [[https://polymake.org/doku.php/release_docs/latest/group.html | documentation]] for more information on the action types. |
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In order to access the complete set of functions dealing with groups, you should switch to the corresponding [[howto/lingo# application| application]]. | In order to access the complete set of functions dealing with groups, you should switch to the corresponding [[user_guide:lingo# application| application]]. |
<code> | <code> |
> application "group"; | > application "group"; |
-2 0 2 1 4 1 1/2-1/2r5 3/2-1/2r5 1/2+1/2r5 3/2+1/2r5 | -2 0 2 1 4 1 1/2-1/2r5 3/2-1/2r5 1/2+1/2r5 3/2+1/2r5 |
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group > print irreducible_decomposition($d->GROUP->MATRIX_ACTION->CHARACTER, $d->GROUP); | group > print irreducible_decomposition($d->GROUP->MATRIX_ACTION->CHARACTER, $d->GROUP); |
polymake: WARNING: available properties insufficient to compute 'CHARACTER_TABLE' | polymake: WARNING: available properties insufficient to compute 'CHARACTER_TABLE' |
</code> | </code> |
4 2 0 1 -1 0 -1 | 4 2 0 1 -1 0 -1 |
1 1 1 1 1 1 1 | 1 1 1 1 1 1 1 |
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| group > $s->REGULAR_REPRESENTATION; |
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group > print $s->REGULAR_REPRESENTATION->CHARACTER; | group > print $s->REGULAR_REPRESENTATION->CHARACTER; |