Abstract

Majorana theory was introduced in 2009 by A.A. Ivanov as an axiomatisation of certain properties of the Monster group $\mathbb{M}$ and the Greiss Algebra $V_{\mathbb{M}}$. The main aim of Majorana theory is to describe the subalgebra structure of $V_{\mathbb{M}}$. Ivanov’s work was inspired by a result of S. Sakuma who classified certain subalgebras of $V_{\mathbb{M}}$ within the context of Vertex Operator Algebras (VOAs), objects which feature in the proof of Montrous Moonshine.

The crucial definition in Majorana theory is that of a Majorana algebra, a commutative real algebra generated by a set of idempotents which obeys certain axioms. Similarly, given a finite group $G$, it is possible to define a Majorana representation of $G$. In this talk, I will cover the basic ideas of Majorana theory before discussing my own work on the Majorana representations of certain groups of interest in the context of the Monster.