Some branching formulas for Kac--Moody Lie algebras

TL;DR

This paper derives branching formulas for fundamental representations of certain Kac--Moody Lie algebras, aiding in understanding their structure and related algebraic objects, and proposes conjectures on generic rings.

Contribution

It introduces new branching rules for Kac--Moody Lie algebras associated with T-shaped graphs and explores their implications for generic rings.

Findings

Derived explicit branching formulas for fundamental representations.

Connected branching rules to generators of generic rings.

Proposed conjectures on the structure of generic rings.

Abstract

In this paper we give some branching rules for the fundamental representations of Kac--Moody Lie algebras associated to $T$-shaped graphs. These formulas are useful to describe generators of the generic rings for free resolutions of length three described in [JWm16]. We also make some conjectures about the generic rings.