A bound on the degree of singular vectors for the exceptional Lie superalgebra $E(5,10)$

TL;DR

This paper establishes upper bounds on the degree of singular vectors in minimal Verma modules for the exceptional Lie superalgebra $E(5,10)$, refining previous estimates through advanced algebraic techniques.

Contribution

It introduces a novel application of Lie pseudoalgebra methods to improve bounds on singular vector degrees in $E(5,10)$ representations.

Findings

Degree of singular vectors is at most 14.

Refined bound shows degree is always ≤12.

Except for finitely many cases, degree ≤10.

Abstract

We use the language of Lie pseudoalgebras to gain information about the representation theory of the simple infinite-dimensional linearly compact Lie superalgebra of exceptional type $E(5,10)$. This technology allows us to prove that the degree of singular vectors in minimal Verma modules is $\leq 14$. A few technical adjustments allow us to refine the bound, proving that the degree must always be $\leq 12$ and it is actually, except for a finite number of cases, $\leq 10$.