Representations of the Exceptional Lie superalgebra $E(3,6): II. Four series of degenerate modules

TL;DR

This paper constructs and analyzes complexes of modules over the exceptional Lie superalgebra E(3,6), providing explicit descriptions of degenerate irreducible modules, their characters, and potential applications to particle physics.

Contribution

It introduces four bigraded complexes with E(3,6) action, classifies their modules, and explicitly constructs all degenerate irreducible E(3,6)-modules, including character computations.

Findings

All images, cokernels, and most kernels are irreducible E(3,6)-modules.

Explicit construction of all degenerate irreducible E(3,6)-modules.

Potential applications to particle physics via symmetry groups.

Abstract

Four $\ZZ_+$-bigraded complexes with the action of the exceptional infinite-dimensional Lie superalgebra E(3,6) are constructed. We show that all the images and cokernels and all but three kernels of the differentials are irreducible E(3,6)-modules. This is based on the list of singular vectors and the calculation of homology of these complexes. As a result, we obtain an explicit construction of all degenerate irreducible E(3,6)-modules and compute their characters and sizes. Since the group of symmetries of the Standard Model $SU(3) \times SU(2) \times U(1)$ (divided by a central subgroup of order six) is a maximal compact subgroup of the group of automorphisms of E(3,6), our results may have applications to particle physics.