A variant of d'Alembert functional equation on monoids

TL;DR

This paper characterizes complex solutions to a generalized d'Alembert functional equation on monoids involving involutive automorphisms and anti-automorphisms, expressing solutions via multiplicative functions and representation characters.

Contribution

It provides a complete description of solutions to a new variant of the d'Alembert functional equation on monoids, linking solutions to representation theory.

Findings

Solutions expressed via multiplicative functions

Solutions involve characters of 2D irreducible representations

Characterization applies to monoids with involutive automorphisms and anti-automorphisms

Abstract

In this paper, we determine the complex-valued solutions of the functional equation $$ f(x\sigma(y))+f(\tau(y)x)=2f(x)f(y)$$ for all $x,y \in M$, where $M$ is a monoid, $\sigma$: $M\longrightarrow M$ is an involutive automorphism and $\tau$: $M\longrightarrow M$ is an involutive anti-automorphism. The solutions are expressed in terms of multiplicative functions, and characters of $2$-dimensional irreducible representations of $M$.