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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| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
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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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | 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 $R^\varphi.$ |
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| DOI: | 10.3842/umzh.v76i12.7976 |