7 Characters and Class Functions

Abstract

In this chapter, we work exclusively over \(\mathbb{C}\), although most of the results hold in greater generality (cf. [MQS15], where the theory is worked out over an arbitrary field). We study the ring \(\mathop{\mathrm{Cl}}\nolimits (M)\) of class functions on a finite monoid M. It turns out that \(\mathop{\mathrm{Cl}}\nolimits (M)\cong \mathbb{C} \otimes _{\mathbb{Z}}\mathop{ G_{0}}\nolimits (\mathbb{C}M)\). The character table of a monoid is defined and shown to be invertible. In fact, it is block upper triangular with group character tables on the diagonal blocks. Inverting the character table allows us to determine, in principle, the composition factors of a representation directly from its character. The fundamental results of this chapter are due to McAlister [McA72] and, independently, to Rhodes and Zalcstein [RZ91].