Webification of symmetry classes of plane partitions

Webs are graphical objects that give a tangible, combinatorial way to compute and classify tensor invariants. Recently, [Gaetz, Pechenik, Pfannerer, Striker, Swanson 2023+] found a rotation-invariant web basis for $\mathrm{SL}_4$, as well as its quantum deformation $U_q(\mathfrak{sl}_4)$, and a bijection between move equivalence classes of $U_q(\mathfrak{sl}_4)$-webs and fluctuating tableaux such that web rotation corresponds to tableau promotion. They also found a bijection between the set of plane partitions in an $a\times b\times c$ box and a benzene move equivalence class of $U_q(\mathfrak{sl}_4)$-webs by determining the corresponding oscillating tableau. In this paper, we similarly find the oscillating tableaux corresponding to plane partitions in certain symmetry classes. We furthermore show that there is a projection from $U_q(\mathfrak{sl}_4)$ invariants to $U_q(\mathfrak{sl}_r)$ for $r=2,3$ for webs arising from certain symmetry classes.