A Combinatorial Approach to Root Multiplicities of a Special Type Rank 3 Kac-Moody Algebras

TL;DR

This paper develops a combinatorial method to compute the dimensions of root spaces in a specific class of rank 3 Kac-Moody algebras, providing explicit formulas based on root properties.

Contribution

It introduces standard form elements and proves they span root spaces, enabling explicit dimension calculations for a special type of rank 3 Kac-Moody algebra.

Findings

Derived a formula for root space dimensions

Introduced standard form elements for root space basis

Provided combinatorial approach for root multiplicities

Abstract

In this paper, we calculate the dimension of root spaces $\mathfrak{g}_{\lambda}$ of a special type rank $3$ Kac-Moody algebras $\mathfrak{g}$. We first introduce a special type of elements in $\mathfrak{g}$, which we call elements in standard form. Then, we prove that any root space is spanned by these elements. By calculating the number of linearly independent elements in standard form, we obtain a formula for the dimension of root spaces $\mathfrak{g}_{\lambda}$, which depends on the root $\lambda$.