Decomposition numbers for Hecke algebras of type \(G(r,p,n)\): the \((\varepsilon,q)\)-separated case. (Q2888908)

In the paper under review, the authors study the modular representations of the cyclotomic Hecke algebras of type \(G(r,p,n)\) when the parameters are \((\varepsilon,q)\)-separated. This assumption concerning the parameters is analogous to the situation in Clifford theory where one commonly assumes that the characteristic \(p\) of the field does not divide \([G(r,1,n):G(r,p,n)]\).NEWLINENEWLINE Here the authors present an algorithm to compute the decomposition numbers of these algebras by exploiting a relationship that exists between them and the decomposition numbers of the related cyclotomic Hecke algebras of type \(G(s,1,m)\), where \(1\leq s\leq r\) and \(1\leq m\leq n\). In addition, the authors develop a Specht module theory for these algebras, explicitly construct their simple modules, and introduce and study analogs of the cyclotomic Schur algebras of type \(G(r,p,n)\) when the parameters are \((\varepsilon,q)\)-separated.NEWLINENEWLINE The main results in this paper rely on two Morita equivalences: the first equivalence reduces the calculation of all decomposition numbers to the case of \(l\)-splittable decomposition numbers, and the second equivalence allows the authors to compute these decomposition numbers using an analog of the cyclotomic Schur algebras for the Hecke algebras of type \(G(r,p,n)\).