# Star-product formulas on finite groups

### Abstract

Given a finite group *G*, associative and noncommutative products * will be constructed on the function space *L*²(*G*), where these star-products are parametrized by the unitary dual *Ĝ* of *G*. If *R _{f}*:

*L*²(

*G*)→

*L*²(

*G*) is the convolution operator

*R*(

_{f}*f*

_{1})=

*f**

*f*

_{1}, then the projections of

*R*on the irreducible spaces of the elements of

_{f}*Ĝ*is equivalent to

*R*

_{f}_{*},

*R*

_{f}_{*}(

*f*

_{1})=

*f**

*f*

_{1}. It is thus natural to consider Harmonic Analysis/Quantum Mechanics on

*L*²(

*G*) using the star-product. In this work, we establish star-product formulas on finite groups, both in the position and momentum representation.

*Proceedings of the Samahang Pisika ng Pilipinas*

**37**, SPP-2019-2D-04 (2019). URL: https://paperview.spp-online.org/proceedings/article/view/SPP-2019-2D-04.