Investigating root multiplicities in the indefinite Kac–Moody algebra $E_{10}$

Abstract

Following a procedure outlined by Kang, we view the generalized eigenspaces, known as root spaces, of the infinite dimensional Kac–Moody algebra $E_{10}$ as generalized eigenspaces for representations of the finite dimensional special linear algebra $A_9$. Then, using the combinatorial representation theory of the special linear Lie algebras, we determine the dimensions of certain root spaces in $E_{10}$.