  We raise the question of whether (a slightly generalized notion of)
$qq$-characters can be constructed purely representation-theoretically. In the
main example of the quantum toroidal $\mathfrak{gl}_1$ algebra, geometric
engineering of adjoint matter produces an explicit vertex operator
$\mathsf{RR}$ which computes certain $qq$-characters, namely Hirzebruch
$\chi_y$-genera, completely analogously to how the R-matrix $\mathsf{R}$
computes $q$-characters. We give a geometric proof of the independence of
preferred direction for the refined vertex in this and more general non-toric
settings.
