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<p>Let <i>n</i> be a positive integer, let <i>R</i> be a (unitary associative) ring, and let <i>M</i><sub><i>n</i></sub>(<i>R</i>) be the ring of all <i>n</i> by <i>n</i> matrices over <i>R.</i> For a permutation <i>σ</i> in the symmetry group Σ<sub><i>n</i></sub> and a ring automorphism <i>φ</i> of <i>R,</i> we introduce the definition of <i>σ</i> - <i>φ</i> permutation matrices. The set <i>B</i><sub><i>n</i></sub>(<i>σ, φ, R</i>) of all <i>σ</i> - <i>φ</i> permutation matrices is proved to be a subring of <i>M</i><sub><i>n</i></sub>(<i>R</i>)<i>.</i> It is shown that the extension <i>B</i><sub><i>n</i></sub>(<i>σ, φ, R</i>) ⊆ <i>M</i><sub><i>n</i></sub>(<i>R</i>) is a separable Frobenius extension. Moreover, if <i>R</i> is a commutative cellular algebra over the invariant subring <i>R</i><sup><i>φ</i></sup> of <i>R,</i> then <i>B</i><sub><i>n</i></sub>(<i>σ, φ, R</i>) is also a cellular algebra over <i>R</i><sup><i>φ</i></sup><i>.</i></p>

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Cellular Algebras and Frobenius Extensions Arising from Two-Parameter Permutation Matrices

  • 期刊论文

  • 发表日期：

    2025年5月1日

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  • Houzhi He,
  • Huabo Xu

摘要

Let n be a positive integer, let R be a (unitary associative) ring, and let Mn(R) be the ring of all n by n matrices over R. For a permutation σ in the symmetry group Σn and a ring automorphism φ of R, we introduce the definition of σ - φ permutation matrices. The set Bn(σ, φ, R) of all σ - φ permutation matrices is proved to be a subring of Mn(R). It is shown that the extension Bn(σ, φ, R) ⊆ Mn(R) is a separable Frobenius extension. Moreover, if R is a commutative cellular algebra over the invariant subring Rφ of R, then Bn(σ, φ, R) is also a cellular algebra over Rφ.