Theorem.The Fourier transform F simultaneously decomposes the left and right regular representations L and R into their irreducible components. Proof.We verify for left regular representation,and the case for R is similar. ix)=FGL(x)F毫=∑创 y∈G da =ΣΣΣ Ndad yeG.g'∈Gj,k='k'=1 G X o(xy)j.ko'(y).j.kXo'.j'.k'l 足Σ空空 Ndoda yeG o.o'EGj.k.t=1 j'.k'=1 IG] Xa(x)j.co(y)eko'y).j.k)o'.j'.k'l =ΣΣ a(x)j.elo.j.k)(o.e.kl cEGj.k.(=1 =⊕[x)⑧1a,] geG The identity Eq.(1)also implies the following basic fact. Fact.IGI=∑pecd 4.Abelian groups For a cyclic group Zn,the irreps are py:Ix)wy for ye[n].The Fourier transform is pe:lx)H斤Zyo'lbW.Fora finite Abelian group,suppsesmicto,×…×Zne,then the irreps arewhere and the Fouriertransform is Pyiy:x1.x) yh…ye For Abelian groups,all irreps are one-dimensional.The converse is also true:Any non-Abelian group has an irrep with degree strictly larger than 1.Theorem. The Fourier transform 𝐹𝐺 simultaneously decomposes the left and right regular representations 𝐿 and 𝑅 into their irreducible components. Proof. We verify for left regular representation, and the case for 𝑅 is similar. The identity Eq.(1) also implies the following basic fact. Fact. |𝐺| = ∑ 𝑑𝜌 2 𝜌∈𝐺̂ . 4. Abelian groups For a cyclic group ℤ𝑛, the irreps are 𝜌𝑦 :|𝑥〉 ↦ 𝜔𝑛 𝑥𝑦 for 𝑦 ∈ [𝑛]. The Fourier transform is 𝜌𝑘:|𝑥〉 ↦ 1 √𝑛 ∑ 𝜔𝑛 𝑥𝑦 𝑦 |𝑦〉. For a finite Abelian group, suppose it is isomorphic to ℤ𝑛1 × …× ℤ𝑛𝑡 , then the irreps are 𝜌𝑦1…𝑦𝑡 :|𝑥1… 𝑥𝑡 〉 ↦ 𝜔𝑛1 𝑥1𝑦1 … 𝜔𝑛𝑡 𝑥𝑡𝑦𝑡 where 𝑥𝑖 ,𝑦𝑖 ∈ [𝑛𝑖 ], and the Fourier transform is 𝜌𝑦1…𝑦𝑡 :|𝑥1…𝑥𝑡 〉 ↦ 1 √𝑛 ∑ 𝜔𝑛1 𝑥1𝑦1 …𝜔𝑛𝑡 𝑥𝑡𝑦𝑡 |𝑦1 … 𝑦𝑡 〉 𝑦1…𝑦𝑡 . For Abelian groups, all irreps are one-dimensional. The converse is also true: Any non-Abelian group has an irrep with degree strictly larger than 1