Hecke Algebras of TypeDnat Roots of Unity

This paper deals with representations of Hecke algebras of typeDn(i.e., q(Dn)) over any fieldKand with any parameterq. Our method is to study the restrictions (to q(Dn)) of the Specht moduleS̃λand simple moduleD̃λof q, 1(Bn). We prove thatS̃λ q(Dn) S̃λ̂ q(Dn), whereλ̂ = (λ(2), λ(1)). When λ(1) = λ(2), we show thatS̃λis a direct sum of its two q(Dn)-submodules with the same dimensions. Via this we give a crude presentations of all the simple modules of q(Dn). When q(Dn) is semisimple, we give a complete set of pairwise non-isomorphic simple modules. Finally, we also give an attempt of using “Murphy basis” philosophy and construct a basis for q(Dn) whennis odd.