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

Abstract

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 \(1/\chi(1)\), 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.