Cellular algebras and Frobenius extensions arising from two-parameter permutation matrices
UDC 512.5 Let $n$ be a positive integer,  let $R$ be a (unitary associative) ring, and let $M_n(R)$ be the ring of all $n$ by $n$ matrices over $R.$  For a permutation $\sigma$ in the symmetry group $\Sigma_n$ and a ring automorphism $\varphi$ of $R,$&n...
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| Date: | 2025 |
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Institute of Mathematics, NAS of Ukraine
2025
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7976 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512836344610816 |
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| author | He, Houzhi Xu, Huabo He, Houzhi Xu, Huabo |
| author_facet | He, Houzhi Xu, Huabo He, Houzhi Xu, Huabo |
| author_sort | He, Houzhi |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
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| datestamp_date | 2025-08-28T14:54:52Z |
| description | UDC 512.5
Let $n$ be a positive integer,  let $R$ be a (unitary associative) ring, and let $M_n(R)$ be the ring of all $n$ by $n$ matrices over $R.$  For a permutation $\sigma$ in the symmetry group $\Sigma_n$ and a ring automorphism $\varphi$ of $R,$  we introduce the definition of $\sigma$-$\varphi$ permutation matrices. The set $B_n(\sigma, \varphi, R)$ of all $\sigma$-$\varphi$ permutation matrices is proved to be a subring of $M_n(R).$  We show that the extension $B_n(\sigma, \varphi, R) \subseteq M_n(R)$ is a separable Frobenius extension. Moreover, if $R$ is a commutative cellular algebra over the invariant subring $R^\varphi$ of $R,$ then $B_n(\sigma, \varphi, R)$ is also a cellular algebra over $R^\varphi.$ |
| doi_str_mv | 10.3842/umzh.v76i12.7976 |
| first_indexed | 2026-03-24T03:35:07Z |
| format | Article |
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| id | umjimathkievua-article-7976 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:35:07Z |
| publishDate | 2025 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-79762025-08-28T14:54:52Z Cellular algebras and Frobenius extensions arising from two-parameter permutation matrices Cellular algebras and Frobenius extensions arising from two-parameter permutation matrices He, Houzhi Xu, Huabo He, Houzhi Xu, Huabo Invariant subring; $\sigma$-$\varphi$ permutation matrix; Cellular algebra UDC 512.5 Let $n$ be a positive integer,  let $R$ be a (unitary associative) ring, and let $M_n(R)$ be the ring of all $n$ by $n$ matrices over $R.$  For a permutation $\sigma$ in the symmetry group $\Sigma_n$ and a ring automorphism $\varphi$ of $R,$  we introduce the definition of $\sigma$-$\varphi$ permutation matrices. The set $B_n(\sigma, \varphi, R)$ of all $\sigma$-$\varphi$ permutation matrices is proved to be a subring of $M_n(R).$  We show that the extension $B_n(\sigma, \varphi, R) \subseteq M_n(R)$ is a separable Frobenius extension. Moreover, if $R$ is a commutative cellular algebra over the invariant subring $R^\varphi$ of $R,$ then $B_n(\sigma, \varphi, R)$ is also a cellular algebra over $R^\varphi.$ УДК 512.5 Клітинні алгебри та розширення Фробеніуса, що виникають iз двопараметричних матриць перестановок Нехай $n$ – натуральне число, $R$ – (унітарне асоціативне) кільце, а $M_n(R)$ – кільце  всіх $n \times n$ матриць над $R.$ Для перестановки $\sigma$ у групі симетрії $\Sigma_n$ і кільцевого автоморфізму $\varphi$ над $R$ введено означення матриць перестановки $\sigma$-$\varphi.$ Доведено, що множина $B_n(\sigma, \varphi, R)$ усіх матриць перестановок $\sigma$-$\varphi$ є підкільцем $M_n(R).$ Показано, що розширення $B_n(\sigma, \varphi, R) \subseteq M_n(R)$ є сепарабельним розширенням Фробеніуса. Навіть більше, якщо $R$ є комутативною клітинною алгеброю над інваріантною підгрупою $R^\varphi$ з $R,$ то $B_n(\sigma, \varphi, R)$ також є клітинною алгеброю над $R^\varphi.$  Institute of Mathematics, NAS of Ukraine 2025-08-27 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7976 10.3842/umzh.v76i12.7976 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 12 (2024); 1838–1850 Український математичний журнал; Том 76 № 12 (2024); 1838–1850 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7976/10290 Copyright (c) 2024 houzhi he, huabo xu |
| spellingShingle | He, Houzhi Xu, Huabo He, Houzhi Xu, Huabo Cellular algebras and Frobenius extensions arising from two-parameter permutation matrices |
| title | Cellular algebras and Frobenius extensions arising from two-parameter permutation matrices |
| title_alt | Cellular algebras and Frobenius extensions arising from two-parameter permutation matrices |
| title_full | Cellular algebras and Frobenius extensions arising from two-parameter permutation matrices |
| title_fullStr | Cellular algebras and Frobenius extensions arising from two-parameter permutation matrices |
| title_full_unstemmed | Cellular algebras and Frobenius extensions arising from two-parameter permutation matrices |
| title_short | Cellular algebras and Frobenius extensions arising from two-parameter permutation matrices |
| title_sort | cellular algebras and frobenius extensions arising from two-parameter permutation matrices |
| topic_facet | Invariant subring $\sigma$-$\varphi$ permutation matrix Cellular algebra |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7976 |
| work_keys_str_mv | AT hehouzhi cellularalgebrasandfrobeniusextensionsarisingfromtwoparameterpermutationmatrices AT xuhuabo cellularalgebrasandfrobeniusextensionsarisingfromtwoparameterpermutationmatrices AT hehouzhi cellularalgebrasandfrobeniusextensionsarisingfromtwoparameterpermutationmatrices AT xuhuabo cellularalgebrasandfrobeniusextensionsarisingfromtwoparameterpermutationmatrices |