Let G be a finite group, $|$G$|<\backslash$infty $,$ and let LG be the vector space of complex functions f$($g$)$

on G$,$ f : G $->$ C$.$ With respect to the inner product

$$f$1|$f$2=$

$1$

$|$G$|$

X

g in G

f$1($g$)$f$2($g$),$f$1,$ f$2$ in LG $,$

LG is a finite$-$dimensional Hilbert space.

$($a$)$ Given a group element h in G$,$ associate with it a linear operator TL : LG $->$ LG defined by

TL : f$($g$)->$

h

TL$($h$)$f

i

$($g$)=$ f$($h

$1$

g$).$

Demonstrate that TL is a unitary representation of G$.$ It is the left regular representation of G$.$

$($b$)$ Given a group element h in G$,$ associate with it a linear operator TR : LG $->$ LG defined by

TR : f$($g$)->$

h

TR$($h$)$f

i

$($g$)=$ f$($gh$).$

Demonstrate that TR is a unitary representation of G$,$ the right regular representation of G$.$

$($c$)$ Prove that TL and TR are equivalent representations.

Hint: Consider the map e: LG $->$ LG defined by f$($g$)->$ fe$($g$)=$ f$($g

$1$

$).$

$($d$)$ For the alternating group A$3$ S$3,$ decompose its regular representation into irreducible

ones