Webification of symmetry classes of plane partitions

TL;DR

This paper extends the web-based combinatorial framework for classifying tensor invariants to symmetry classes of plane partitions, establishing new bijections and projections among webs, oscillating tableaux, and plane partitions.

Contribution

It introduces web representations for symmetry classes of plane partitions and demonstrates projections from $U_q(rak{sl}_4)$ invariants to lower-rank cases.

Findings

Bijections between symmetry class plane partitions and oscillating tableaux.

Projection from $U_q(rak{sl}_4)$ invariants to $U_q(rak{sl}_r)$ for $r=2,3$.

Web representations for symmetry classes of plane partitions.

Abstract

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.