Fast Generalized Fourier Transforms

Abstract Let G be a f inite group. Then Ls(G), the linear complexity of a suitable Wedderburn transform corresponding to G, is smaller than 2·|G|2. We improve this trivial upper bound by showing that Ls(G) is smaller than min {(s( T )−l( T ))·|G|+7 q( T ) ·|G| 3 2 , where the minimum is taken over all towers T =(G n >G n−1 >⋯>G 0 ={1}) of subgroups of G=Gn, and where q( T ) (resp. s( T )) is the maximum (resp. sum) of all indices [Gi+1:Gi] corresponding to T and l( T )=n is the length of the tower. A similar upper bound is valid for the linear complexity of the inverse of such a Wedderburn transform. For symmetric groups our technique yields the stronger estimate Ls(Sn)=0(|Sn|·log3|Sn|).