
TL;DR
This paper derives difference equations for signature characters of highest weight representations of finite-dimensional Lie algebras, specifically A_2 and B_2, revealing patterns similar to those in Virasoro algebra representations.
Contribution
It introduces a novel approach using difference equations to compute signature characters for A_2 and B_2 Lie algebra representations, extending known signature patterns.
Findings
Derived difference equations for signature characters
Computed signature characters for A_2 and B_2
Revealed patterns analogous to Virasoro algebra
Abstract
The signatures of the inner product matrices on a Lie algebra's highest weight representation are encoded in the representation's signature character. We show that the signature characters of a finite-dimensional Lie algebra's highest weight representations obey simple difference equations that have a unique solution once appropriate boundary conditions are imposed. We use these results to derive the signature characters of all and highest weight representations. Our results extend, and explain, signature patterns analogous to those observed by Friedan, Qiu and Shenker in the Virasoro algebra's representation theory.
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