representation is defined by for The right-regular representation is defined analogously by for Representations, modules and the convolution algebra Let be a finite group, let be a commutative ring and let be the group algebra of over This algebra is free and a basis can be indexed by the elements of Most often the basis is identified with . Every element can then be uniquely expressed as with . The multiplication in extends that in distributively. Now let be a –module and let be a linear representation of in We define for all and . By linear extension is endowed with the structure of a left-–module. Vice versa we obtain a linear representation of starting from a –module . Additionally, homomorphisms of representations are in bijective correspondence with group algebra homomorphisms. Therefore, these terms may be used interchangeably. This is an example of an isomorphism of categories. Suppose In this case the left –module given by itself corresponds to the left-regular representation. In the same way as a right –module corresponds to the right-regular representation. In the following we will define the convolution algebra: Let be a group, the set is a –vector space with the operations addition and scalar multiplication then this vector space is isomorphic to The convolution of two elements defined by makes an algebra. The algebra is called the convolution algebra. The convolution algebra is free and has a basis indexed by the group elements: where Using the properties of the convolution we obtain: We define a map between and by defining on the basis and extending it linearly. Obviously the prior map is bijective. A closer inspection of the convolution of two basis elements as shown in the equation above reveals that the multiplication in corresponds to that in Thus, the convolution algebra and the group algebra are isomorphic as algebras. The involution turns into a –algebra. We have A representation of a group extends to a –algebra homomorphism by Since multiplicity is a characteristic property of algebra homomorphisms, satisfies If is unitary, we also obtain For the definition of a unitary representation, please refer to the chapter on properties. In that chapter we will see that (without loss of generality) every linear representation can be assumed to be unitary. Using the convolution algebra we can implement a Fourier transformation on a group In the area of harmonic analysis it is shown that the following