The structure over \(\mathbb{Z}[q,q^{-1}]\) of Hecke orders of dihedral groups
The article provides an inductive description of Hecke orders \({\mathcal H}_G\) over \(\mathbb{Z}[q,q^{-1}]\), where \(G\) is a dihedral group of order \(2\cdot p^n\), and \(p\) is an odd prime. Thus \({\mathcal H}_G\) is an order in the separable \(\mathbb{Q}(q)\)-algebra \(\mathbb{Q}(q)\otimes_{\mathbb{Z}[q]}{\mathcal H}_G\cong\mathbb{Q}(q)[G]\). Notice that the two-dimensional coefficient ring \(\mathbb{Z}[q,q^{-1}]\) belongs to an integral affine scheme \(X\) that involves all algebraic number fields as well as the function fields \(\mathbb{F}_p(q)\) for any rational prime \(p\). Namely, these fields arise as function fields of irreducible closed subschemes of \(X\). Therefore, the representation theory over \(\mathbb{Z}[q,q^{-1}]\) comprises the theory of (integral and rational) representations over algebraic number fields as well as modular representation theory as a special case. Let \(\vartheta_i\) be a primitive \(p^i\)-th root of unity, and let \(K_i\) denote the quotient field of \(R_i:=\mathbb{Z}[q,q^{-1}]\otimes\mathbb{Z}[\eta_i]\), where \(\eta_i:=\vartheta_i+\vartheta^{-1}_i\). Then \(\mathbb{Q}(q)[G]\) has \(n\) irreducible representations of the form \(M_2(K_i)\). By a suitable choice of the basis in \(M_2(K_i)\), the author shows that the projection of \({\mathcal H}_G\) into \(M_2(K_i)\) is of the form \(\left(\begin{smallmatrix} R_i &\pi\\ R_i &R_i\end{smallmatrix}\right)\), where \(\pi_i\) is the principal ideal generated by \(q^2-\eta_iq+1\). (Hence \(R_i/\pi_i\cong\mathbb{Z}[\vartheta_i]\).) The order \({\mathcal H}_G\) is then constructed inductively from these projections by a series of amalgamations.
scientific article; zbMATH DE number 1428246