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 $R,$ 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 $R^\varphi.$