{"id":30136630,"url":"https://github.com/dylanjohnston/fpa5-lifting-idempotents","last_synced_at":"2025-08-10T23:10:26.405Z","repository":{"id":307238200,"uuid":"1024484092","full_name":"DylanJohnston/FpA5-lifting-idempotents","owner":"DylanJohnston","description":"arXiv:2507.21316 Notebooks showing the lifting of primitive idempotents from the group algebras F_pA_5 (p=2,3,5) to the group algebra QA_5.","archived":false,"fork":false,"pushed_at":"2025-07-30T05:49:47.000Z","size":1392,"stargazers_count":0,"open_issues_count":0,"forks_count":0,"subscribers_count":0,"default_branch":"main","last_synced_at":"2025-07-30T07:32:48.755Z","etag":null,"topics":["idempotent","lifting","mathematics","p-adic-numbers","representation-theory","sagemath","semiperfect","symmetric-group"],"latest_commit_sha":null,"homepage":"","language":"Jupyter Notebook","has_issues":true,"has_wiki":null,"has_pages":null,"mirror_url":null,"source_name":null,"license":"gpl-3.0","status":null,"scm":"git","pull_requests_enabled":true,"icon_url":"https://github.com/DylanJohnston.png","metadata":{"files":{"readme":"README.md","changelog":null,"contributing":null,"funding":null,"license":"LICENSE","code_of_conduct":null,"threat_model":null,"audit":null,"citation":null,"codeowners":null,"security":null,"support":null,"governance":null,"roadmap":null,"authors":null,"dei":null,"publiccode":null,"codemeta":null,"zenodo":null}},"created_at":"2025-07-22T19:15:56.000Z","updated_at":"2025-07-30T05:49:51.000Z","dependencies_parsed_at":"2025-07-30T07:32:55.062Z","dependency_job_id":"e0c4cdf7-11f8-400f-8e53-c882cc53f0fc","html_url":"https://github.com/DylanJohnston/FpA5-lifting-idempotents","commit_stats":null,"previous_names":["dylanjohnston/fpa5-lifting-idempotents"],"tags_count":null,"template":false,"template_full_name":null,"purl":"pkg:github/DylanJohnston/FpA5-lifting-idempotents","repository_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/DylanJohnston%2FFpA5-lifting-idempotents","tags_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/DylanJohnston%2FFpA5-lifting-idempotents/tags","releases_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/DylanJohnston%2FFpA5-lifting-idempotents/releases","manifests_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/DylanJohnston%2FFpA5-lifting-idempotents/manifests","owner_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners/DylanJohnston","download_url":"https://codeload.github.com/DylanJohnston/FpA5-lifting-idempotents/tar.gz/refs/heads/main","sbom_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories/DylanJohnston%2FFpA5-lifting-idempotents/sbom","host":{"name":"GitHub","url":"https://github.com","kind":"github","repositories_count":269803855,"owners_count":24477658,"icon_url":"https://github.com/github.png","version":null,"created_at":"2022-05-30T11:31:42.601Z","updated_at":"2022-07-04T15:15:14.044Z","status":"online","status_checked_at":"2025-08-10T02:00:08.965Z","response_time":71,"last_error":null,"robots_txt_status":"success","robots_txt_updated_at":"2025-07-24T06:49:26.215Z","robots_txt_url":"https://github.com/robots.txt","online":true,"can_crawl_api":true,"host_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub","repositories_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repositories","repository_names_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/repository_names","owners_url":"https://repos.ecosyste.ms/api/v1/hosts/GitHub/owners"}},"keywords":["idempotent","lifting","mathematics","p-adic-numbers","representation-theory","sagemath","semiperfect","symmetric-group"],"created_at":"2025-08-10T23:10:24.056Z","updated_at":"2025-08-10T23:10:26.387Z","avatar_url":"https://github.com/DylanJohnston.png","language":"Jupyter Notebook","funding_links":[],"categories":[],"sub_categories":[],"readme":"# Lifting Idempotents from $`\\mathbb{F}_pA_5`$ to $`\\mathbb{Z}_{(p)}A_5`$\n\nSupplementary code for the paper \"[On a question by Roggenkamp about group algebras](https://arxiv.org/abs/2507.21316)\", which was joint work with Dmitriy Rumynin.\n\nLet $`\\mathbb{F}_p`$ be the prime field of characteristic $`p`$, and let $`\\mathbb{Z}_{(p)}`$ denote the localisation of the integers at the prime $`p`$. Let $`A_5`$ be the alternating group on $`5`$ letters. We have $`|A_5| = 60 = 2^2\\cdot3\\cdot5`$.\n\nLifting an idempotent $`\\overline{e} \\in \\mathbb{F}_pA_5`$ to $`\\mathbb{Z}_{(p)}A_5`$ amounts to finding an element $`e \\in \\mathbb{Z}_{(p)}A_5`$ such that:\n- $`e \\equiv \\overline{e} \\pmod{p}`$\n- $`e^2 = e`$\n\nIn this repository, primitive idempotents (equivalently, indecomposable projective modules) are lifted from $`\\mathbb{F}_pA_5`$ to $`\\mathbb{Z}_{(p)}A_5`$ for the \"modular primes\" $`p = 2`$, $`3`$, and $`5`$.\n\nFor each of these primes, there are three indecomposable projective modules to lift. The process of lifting them is as follows:\n- Identify primitive idempotents in $`\\mathbb{F}_pA_5`$,\n- Lift these primitive idempotents to the group ring over the $`p`$-adic integers $`\\widehat{\\mathbb{Z}_p}A_5`$.\n- Check that all coefficients in the lift to $`\\widehat{\\mathbb{Z}_p}A_5`$ have $`p`$-adic expansions with repeating digits, i.e., the coefficients lie in $`\\mathbb{Z}_{(p)} \\subset \\mathbb{Q}`$.\n\nDetails can be found in the following notebooks:\n- [idempotent-lift-F_2A_5.ipynb](https://github.com/DylanJohnston/FpA5-lifting-idempotents/blob/main/idempotent-lift-F_2A_5.ipynb)\n- [idempotent-lift-F_3A_5.ipynb](https://github.com/DylanJohnston/FpA5-lifting-idempotents/blob/main/idempotent-lift-F_3A_5.ipynb)\n- [idempotent-lift-F_5A_5.ipynb](https://github.com/DylanJohnston/FpA5-lifting-idempotents/blob/main/idempotent-lift-F_5A_5.ipynb)\n\nAll three notebooks follow a similar structure. In fact, the $`p = 3`$ case was completed first, and the others are copies of it with the relevant modifications. This approach results in heavy code repetition, but in return, each notebook is fully self-contained. This was deemed to be worth it.\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fdylanjohnston%2Ffpa5-lifting-idempotents","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Fdylanjohnston%2Ffpa5-lifting-idempotents","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Fdylanjohnston%2Ffpa5-lifting-idempotents/lists"}