Opened 11 years ago

Last modified 10 years ago

## #10052 closed enhancement

# Improve the implementation of the Steenrod algebra — at Initial Version

Reported by: | jhpalmieri | Owned by: | AlexGhitza |
---|---|---|---|

Priority: | major | Milestone: | sage-4.7.1 |

Component: | algebra | Keywords: | steenrod, notebook |

Cc: | jvkersch, niles | Merged in: | |

Authors: | John Palmieri | Reviewers: | |

Report Upstream: | N/A | Work issues: | |

Branch: | Commit: | ||

Dependencies: | Stopgaps: |

### Description

The attached patch does several things:

- it moves the Steenrod algebra code to a subdirectory of algebras. For one thing, there are already 4 files, and for another, I hope that more will be added: several people are working on related projects.

- it reimplements the Steenrod algebra using
`CombinatorialFreeModule`

, which provides a number of conveniences: scalar multiplication is already defined, as are tensor products, etc. Then for example the antipode or the coproduct can be defined just on basis elements, and the linear extension to all elements is handled automatically.

- it implements another way of computing products, using admissible sequences and the Adem relations. This provides a good way of checking for bugs: with two completely different algorithms for computing products, one can compute the same product two ways and compare answers.

- it implements sub-Hopf algebras of the Steenrod algebra. These were classified 35 years ago, and for some applications people want to use sub-Hopf algebras rather than the whole thing.

- the
`TestSuite`

has been improved: all components now pass, whereas before, we had some failures. From the*old*steenrod_algebra.py:sage: TestSuite(A).run() # todo: fix category inheritance for elements of A Failure in _test_category: ... ------------------------------------------------------------ The following tests failed: _test_category Failure in _test_elements The following tests failed: _test_elements

From the new one:sage: TestSuite(SteenrodAlgebra()).run() sage: TestSuite(SteenrodAlgebra(profile=[4,3,2,2,1])).run() sage: TestSuite(SteenrodAlgebra(basis='adem')).run() sage: TestSuite(SteenrodAlgebra(basis='wall')).run() sage: TestSuite(SteenrodAlgebra(basis='arnonc')).run() # long time sage: TestSuite(SteenrodAlgebra(basis='woody')).run() # long time sage: A3 = SteenrodAlgebra(3) sage: A3.category() Category of graded hopf algebras with basis over Finite Field of size 3 sage: TestSuite(A3).run() sage: TestSuite(SteenrodAlgebra(basis='adem', p=3)).run() sage: TestSuite(SteenrodAlgebra(basis='pst_llex', p=7)).run() # long time sage: TestSuite(SteenrodAlgebra(basis='comm_deg', p=5)).run() # long time

Not only are there no failures, but there are many more executions of the suite. This yields some repetition, but the method`an_element`

varies depending on the values of`basis`

and`p`

, so there are also new tests run with each execution.

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patch for sagenb repo