Proof of Kac and Rudakov's Conjecture on Generalized Verma Module over Lie Superalgebra E(5,10)

TL;DR

This paper proves Kac and Rudakov's conjecture on the classification of degenerate generalized Verma modules over the Lie superalgebra E(5,10), which is linked to the symmetries of the SU(5) Grand Unified Model.

Contribution

It provides a complete proof of the conjecture and explicitly constructs all nontrivial singular vectors for E(5,10).

Findings

Classification of all degenerate generalized Verma modules over E(5,10)

Explicit construction of singular vectors degree by degree

Insights into the representation theory relevant to SU(5) GUT

Abstract

The exceptional infinite-dimensional linearly compact simple Lie superalgebra ${E}(5,10)$, which Kac believes, is the algebra of symmetries of the ${SU}_{5}$ Grand Unified Model. In this paper, we give a proof of Kac and Rudakov's conjecture about the classification of all the degenerate generalized Verma module over ${E}(5,10)$. Also, we work out all the nontrivial singular vectors degree by degree. It is a potential that the representation theory of ${E}(5,10)$ will shed new light on various features of the the ${SU}_{5}$ Grand unified model.