Define the discrete symmetry operator acting on second-kind boson basis vectors

Determine and formalize the discrete symmetry operator that acts on the second-kind even bosonic basis vectors A^{II}_{f} (the boson “basis vectors” that transform a given family member across families), and characterize its transformation properties and algebraic action on these basis vectors in the framework presented for even-dimensional spaces (e.g., d = 5 + 1 and d = 13 + 1).

Background

In the paper, two kinds of even bosonic basis vectors are introduced: A^{I}_{f}, which act within a family, and A^{II}_{f}, which map the same family member across different families. Discrete symmetry operations are analyzed for the first kind, while for the second kind the authors state that further work is needed.

The unresolved point concerns specifying the discrete symmetry operator—denoted in the text as acting on the second-kind bosonic basis vectors—and giving its explicit definition and action rules (e.g., how it composes with nilpotents/projectors and transforms labels).