Cellular algebras and Frobenius extensions arising from two-parameter permutation matrices

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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 σ in the symmetry group Σ n and a ring automorphism φ of R , we introduce the definition of σ - φ permutation matrices. The set B n ( σ , φ , R ) of all σ - φ permutation matrices is proved to be a subring of M n ( R ) . We show that the extension B n ( σ , φ , R ) ⊆ M n ( R ) is a separable Frobenius extension. Moreover, if R is a commutative cellular algebra over the invariant subring R φ of R , then B n ( σ , φ , R ) is also a cellular algebra over R φ .