# On the Existence of a Group Orthonormal Basis

### From NZJM

**New Zealand Journal of Mathematics**

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

**Peter Zizler**

Department of Mathematics, Physics and Engineering

Mount Royal University,

Calgary, Canada

**Abstract** Let *G* be a finite group and let *l*^{2}(*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 {α_{i}} in *l*^{2}(*G*) is *G*-orthonormal if

In our paper we prove that *l*^{2}(*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 *l*^{2}(*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.