Matrix units and primitive idempotents for the Hecke algebra of type Dn

Abstract Let n ∈ Z ≥ 4 and H q ( D n ) be the semisimple Hecke algebra of type D n with Hecke parameter q ∈ K × . For each simple H q ( D n ) -module V, we use the Hecke generators of H q ( D n ) to construct explicitly a quasi-idempotent z V (i.e., z V 2 = c V z V for some c V ∈ K × ) which is defined over a natural integral form of H q ( D n ) , such that e V : = c V − 1 z V is a primitive idempotent and e V H q ( D n ) ≅ V as right H q ( D n ) -module. We use the seminormal bases of the Hecke algebra H q ( B n ) of type B n to construct a complete set of pairwise orthogonal primitive idempotents of H q ( D n ) , to obtain an explicit seminormal basis of H q ( D n ) as well as a new seminormal construction for each simple module over H q ( D n ) . As byproducts, we discover some rational property of certain square-roots of quotients of γ-coefficients for H q ( B n ) , which play a key role in the proof of the main results of the paper.