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Webification of symmetry classes of plane partitions

Ashleigh Adams, Jessica Striker

TL;DR

This work develops a lattice-word framework for the $U_q(\mathfrak{sl}_4)$-webs associated with plane partitions, focusing on symmetry classes (SPP, CSPP, TSPP, TSSCPP) and their fundamental domains. By analyzing boundary words, trips, and oscillating tableaux, the authors describe explicit lattice-word descriptions for each symmetry class and count the distinct classes. They then introduce an algorithm that projects $U_q(\mathfrak{sl}_4)$-invariants to $U_q(\mathfrak{sl}_r)$-invariants for $r=2$ or $3$ depending on the symmetry class, yielding decorated non-crossing matchings (for $r=2$) or non-elliptic $U_q(\mathfrak{sl}_3)$-webs (for $r=3$). The paper provides concrete examples mapping plane-partition–web correspondences through the projection, and sets up a framework linking benzene moves, boundary data, and promotion-like dynamics on tableaux. This connects symmetry-restricted plane partitions with representation-theoretic projections, offering a combinatorial bridge between high-rank invariants and lower-rank counterparts with potential dynamical interpretation via promotion.

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.

Webification of symmetry classes of plane partitions

TL;DR

This work develops a lattice-word framework for the -webs associated with plane partitions, focusing on symmetry classes (SPP, CSPP, TSPP, TSSCPP) and their fundamental domains. By analyzing boundary words, trips, and oscillating tableaux, the authors describe explicit lattice-word descriptions for each symmetry class and count the distinct classes. They then introduce an algorithm that projects -invariants to -invariants for or depending on the symmetry class, yielding decorated non-crossing matchings (for ) or non-elliptic -webs (for ). The paper provides concrete examples mapping plane-partition–web correspondences through the projection, and sets up a framework linking benzene moves, boundary data, and promotion-like dynamics on tableaux. This connects symmetry-restricted plane partitions with representation-theoretic projections, offering a combinatorial bridge between high-rank invariants and lower-rank counterparts with potential dynamical interpretation via promotion.

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 , as well as its quantum deformation , and a bijection between move equivalence classes of -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 box and a benzene move equivalence class of -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 invariants to for for webs arising from certain symmetry classes.

Paper Structure

This paper contains 16 sections, 8 theorems, 28 equations, 24 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background
  3. $U_q(\mathfrak{sl}_r)$-webs and their representation theory
  4. Computing an invariant from a web
  5. Web bases
  6. Hourglass plabic graphs
  7. Oscillating tableaux and their lattice words
  8. Symmetry classes of plane partitions
  9. Plane Partitions as webs
  10. A description of hourglass plabic graphs on plane partitions
  11. A description of fundamental domains of plane partitions and the corresponding lattice words
  12. A description of hourglass edges along boundaries of fundamental domains
  13. Proof of lattice words of symmetry classes
  14. A combinatorial map between invariant spaces
  15. Example of Theorem \ref{['thm:big invariant to small invariant']}: $U_q(\sl_4)$ to $U_q(\sl_2)$
  16. ...and 1 more sections

Key Result

Theorem 1.1

The $U_q(\sl_4)$-web equivalence class given by contracted, fully-reduced hourglass plabic graphs with boundary word is in bijection with the set $\mathrm{PP}(a,b,c).$

Figures (24)

  • Figure 1: Examples of plane partitions inside a $4\times 4\times 4$-box, with the symmetry class listed below it and its corresponding fundamental domain. See Section \ref{['subsec: symmetry-classes-of-pp']} for details.
  • Figure 2: On the left we have an hourglass plabic graph, and on the right is the resulting hourglass plabic graph after the application of a benzene move.
  • Figure 3: A totally symmetric self-complementary plane partition in an $8\times 8\times 8$ box. The $U_q(\sl_4)$-web in bijection with this plane partition is drawn on top of the plane partition, restricted to the fundamental domain of the symmetry class. We have labeled the top most face (base face) as $F_0$. The boundary word corresponding to this web (and thus also this plane partition) is: $1~1~1~1~2~\overline{4}~2~\overline{4}~2~\overline{4}~2~(34)~4~4~(34)~4~(34)~(34).$
  • Figure 4: On the left is a $U_q(\sl_4)$-web of type $\underline{c} = ( \textcolor{orange}{1}, \textcolor{orange}{-1}, \textcolor{orange}{1}, \textcolor{orange}{-1}, \textcolor{orange}{1}, \textcolor{orange}{-1}, ).$ with labeled edge weights. On the right, is the same web as an hourglass plabic graph and given a proper coloring.
  • Figure 5: Here we give an example of Definition \ref{['def:oscillating-tab']} by building an oscillating tableau on the word $\omega = 1~\Bar{4}~2~\Bar{2}~4~\Bar{1}.$ The pictorial visualization on top represents the oscillating tableau $T$ given underneath.
  • ...and 19 more figures

Theorems & Definitions (32)

  • Theorem 1.1: gaetz2023rotation
  • Theorem 1.2
  • Definition 2.1: fraser2019dimers
  • Example 2.2
  • Theorem 2.3: gaetz2023rotation
  • Definition 2.4: postnikov2006total
  • Definition 2.5: gaetz2023rotation
  • Definition 2.6: gaetz2023rotation
  • Definition 2.7: gaetz2023rotation
  • Example 2.8
  • ...and 22 more