# On the Existence of a Group Orthonormal Basis

### New Zealand Journal of Mathematics

Vol. 45, (2015), Pages 45-52

Peter Zizler

Department of Mathematics, Physics and Engineering

Mount Royal University,

Abstract Let G be a finite group and let l2(G) be a finite dimensional Hilbert space of all complex valued functions for which the elements of $G$ form the (standard) orthonormal basis. We say a set of functions i} in l2(G) is G-orthonormal if

$\sum_{t \in G} \overline{\alpha}_i(t) \alpha_j(t\tau) = \alpha_i(\tau) \delta_{i,j} \mbox{ for all } \tau \in G.$

In our paper we prove that l2(G) admits a G-orthonormal basis if and only if G is an abelian group. Moreover, if G is non-abelian than the size of the largest G-orthonormal set in l2(G) is the sum of the degrees of the irreducible representations of G.

Keywords G-orthonormal basis, non-abelian Fourier transform, non-abelian convolution, irreducible representation.

Classification (MSC2000) Primary 43A32, Secondary 42C99.