Rotation-invariant web bases from hourglass plabic graphs

TL;DR

The paper constructs the first rotation-invariant web basis for Inv_{U_q(\mathfrak{sl}_4)}(\bigwedge_q^{\underline{c}} V_q) by introducing hourglass plabic graphs, whose move-equivalence classes biject with 4-row fluctuating tableaux via the maps \mathcal{T} and \mathcal{G}. A crystal-theoretic growth algorithm realizes these bases, establishing unitriangularity and a leading-term separation word, and skein-type reductions express arbitrary webs in the basis. The framework unifies existing rotation-invariant bases for r \le 4 and extends naturally to semistandard settings, connecting to the symmetrized six-vertex model and to combinatorial objects like alternating sign matrices and plane partitions. The results yield concrete combinatorial tools for computing tensor invariants, with potential cluster-algebra and geometric-representation theory applications through the established bijections between graphs, tableaux, and growth data.

Abstract

Webs give a diagrammatic calculus for spaces of tensor invariants. We introduce hourglass plabic graphs as a new avatar of webs, and use these to give the first rotation-invariant $U_q(\mathfrak{sl}_4)$-web basis, a long-sought object. The characterization of our basis webs relies on the combinatorics of these new plabic graphs and associated configurations of a symmetrized six-vertex model. We give growth rules, based on a novel crystal-theoretic technique, for generating our basis webs from tableaux and we use skein relations to give an algorithm for expressing arbitrary webs in the basis. We also discuss how previously known rotation-invariant web bases can be unified in our framework of hourglass plabic graphs.

Figures (49)

• Figure 1: A top fully reduced hourglass plabic graph $G$ and its corresponding $4$-row rectangular standard tableau $\mathcal{T}(G)$. The purple ($\textcolor{amethyst}{\blacksquare}$) $\mathop{\mathrm{\mathsf{trip}}}\nolimits_1$-, orange ($\textcolor{amber}{\blacksquare}$) $\mathop{\mathrm{\mathsf{trip}}}\nolimits_2$-, and green ($\textcolor{dark}{\blacksquare}$) $\mathop{\mathrm{\mathsf{trip}}}\nolimits_3$-strands are drawn, showing that $\mathop{\mathrm{\mathsf{trip}}}\nolimits_i(G)(1)=5, 10,$ and $13$ for $i=1,2,$ and $3$, respectively.
• Figure 2: A benzene move (leftmost) and the square moves for hourglass plabic graphs. The color reversals of these moves are also allowed.
• Figure 3: Building blocks of CKM-style webs for $U_q(\mathfrak{sl}_r)$ corresponding to products, coproducts, duals, etc. Multiplicities are in $[r]$, and multiplicities on the rightmost diagrams are omitted. Upward arrows indicate duals.
• Figure 4: Translations between $U_q(\mathfrak{sl}_r)$-webs and CKM-style webs. Since $0$-edges in CKM-style webs correspond to the unit object $\mathbb{C}(q)$, they are typically unwritten.
• Figure 5: Tag sign relations for CKM-style webs and $U_q(\mathfrak{sl}_r)$-webs.

Theorems & Definitions (214)

• Theorem A: See \ref{['thm:web-basis']}
• Theorem B: See \ref{['thm:main-bijection']}
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Example 2.4
• Lemma 2.5
• proof
• Example 2.6
• Definition 2.7