From Characters to Matrices: An Elementary Construction of Irreducible Representations of Finite Groups

Let $G$ be a finite group and let $\chi$ be an ordinary irreducible character. We give an elementary algorithm which constructs explicit matrices affording $\chi$. The regular representation provides a canonical ambient representation, and the usual central idempotent projects onto the $\chi$-isotypic component. The main step is then to maximize the squared norm of a diagonal matrix coefficient on the unit sphere of this component. The maximum is $1\mathrm{/}\chi (1)$, and it is attained precisely by vectors whose cyclic span is an irreducible subrepresentation affording $\chi$. Thus the construction reduces the passage from characters to matrices to a concrete optimization problem. The same extraction method applies inside any smaller ambient representation containing $\chi$, such as an induced representation from a subgroup. We complement the theory with a discussion on dimension reduction, robust numerical implementation, and an explicit $S_4$ example.