the structure to the products of its irreducible representations to construct a four generation lepton model. In Section 3 we work out the lepton mass spectrum and discuss how the tribimaximal mixing is embedded in this model. Some of the rich flavor physics phenomenologies of this model are also discussed. Finally, we conclude in Section 4 with some comments on collider physics implications. We also comment on the natural extension of our $A_5$ model to the double icosahedral group $I'$ as the flavor symmetry to describe four generations of both quarks and leptons.

The model
=========

There are five irreducible representations (irreps) of $A_{5}$: one singlet **1**, two triplets, $\textbf{3}$ and $\textbf{3}'$, one quartet $\textbf{4}$ and one quintet $\textbf{5}$. The character table and the multiplication rules of the five irreducible representations are given in Appendix A.

Our model is based on $A_{5}$ with an additional $Z_{2}\times Z_{3}$ symmetry to avoid unwanted terms in the Lagrangian. We assign the four generations of left-handed lepton doublets to $A_{5}\times Z_{2}\times Z_{3}$ as follows: $$\begin{aligned}
\label{lepdbt}
\underbrace{\left(\begin{array}{c}\nu_{4} \\ l_{4} \end{array}\right)_{L} \;\;
\left(\begin{array}{c}\nu_{3} \\ l_{3} \end{array}\right)_{L} \;\;
\left(\begin{array}{c}\nu_{2} \\ l_{2} \end{array}\right)_{L} \;\;
\left(\begin{array}{c}\nu_{1} \\ l_{1} \end{array}\right)_{L}}_{L_{L4} (\textbf{4}, +1,\omega)} \;\;, \end{aligned}$$ where $\left(\begin{array}{c}\nu_{i} \\ l_{i} \end{array}\right)_{L}$ 