SL4 webs and promotion permutations of random 4-row tableaux     permutations

Tableaux and lattice words. A standard Young tableau $T$ of rectangular shape $4\times c$ is encoded by its lattice word $w=w_1\cdots w_n$, $n=4c$, where $w_k\in\{1,2,3,4\}$ is the row containing $k$. A uniformly random $T$ is sampled with the Greene–Nijenhuis–Wilf hook walk. Their number is the dimension of the $\mathrm{SL}_4$-invariant space $\mathrm{Inv}(V^{\otimes n})$.

Promotion permutations. Promotion $\mathcal P$ removes $1$, slides by jeu de taquin, inserts $n+1$ and subtracts $1$. On lattice words, with $j_0=1$ and $j_i$ the first index $\ge j_{i-1}$ at which the prefix $w_1\cdots w_{j_i}$ has as many letters $i$ as $i+1$, promotion deletes $w_1$, changes $w_{j_i}=i+1$ to $i$ for $i=1,2,3$, and appends $4$. The entry $j_i$ is the one that crosses from row $i+1$ into row $i$. Following [GPPSS, Promotion permutations], $\mathrm{prom}_i(T)(b)\equiv j_i(\mathcal P^{b-1}T)+b-1 \pmod n$. Then $\mathrm{prom}_3=\mathrm{prom}_1^{-1}$, $\mathrm{prom}_2$ is a fixed-point-free involution, the antiexcedances of $\mathrm{prom}_i$ are exactly the entries in rows $1,\ldots,i$ (so $\mathrm{prom}_1,\mathrm{prom}_2$ determine $T$), and $\mathrm{prom}_i(\mathcal P T)=\sigma^{-1}\mathrm{prom}_i(T)\,\sigma$ for the long cycle $\sigma$. The panels show the permutation matrices: row $b$, column $\mathrm{prom}_i(b)$.

Hourglass plabic graphs. An hourglass plabic graph is a bipartite planar graph in a disk with boundary vertices $b_1,\ldots,b_n$ clockwise and all internal vertices of degree $4$ counted with multiplicity; an edge of multiplicity $2$ is an hourglass, twisted so that its two strands have the same clockwise order at both ends. The trip permutation $\mathrm{trip}_a$ follows the strand from $b_i$ taking the $a$-th left turn at white vertices and the $a$-th right turn at black vertices. A graph is fully reduced if no graph in its move class (benzene, square and contraction moves) has a $4$-cycle through an hourglass; square faces of simple edges are allowed, and they are where square moves act. Move classes of fully reduced graphs correspond bijectively to $4$-row rectangular (fluctuating) tableaux with $\mathrm{trip}_\bullet(G)=\mathrm{prom}_\bullet(T)$, so promotion becomes rotation up to moves. The rotation-invariant $\mathrm{SL}_4$ web basis of [GPPSS, Invent. Math.] takes the top graph of each class, the one with no clockwise benzene face; the growth output drawn here is fully reduced and in the right class, but may differ from that basis web by benzene moves. For standard $T$ all boundary vertices are black.

Growth rules. The map $T\mapsto G$ is the growth algorithm of [GPPSS, §5]. Put $n$ dangling strands labeled $w_1,\ldots,w_n$ on a line, pointing down. Repeatedly replace a factor of the word of dangling labels by a crossing ($\mathsf X$) of two adjacent strands or a cap $1\bar 1\to\varnothing$, $\bar 44\to\varnothing$, using the $88$ short rules (length $2$ or $3$; in a rule of length $3$ two adjacent strands cross and the remaining letter, on the left or on the right, is an unchanged witness) and two families of long rules $14uw\to 41uw$ ($u$ any word in $\bar2,\bar3$, $w\in\{\bar1,2,3,4\}$) and $23uw\to\bar1\bar4uw$ ($u$ any word in $2,3$, $w\in\{\bar1,\bar2,\bar3,4\}$), with their images under the symmetries, which put the witnesses on the left. Barred labels are strands pointing up. Every sequence of choices terminates. The result is a symmetrized six-vertex configuration: every crossing is a sink, a source, or transmitting (two adjacent edges in, two out). Sinks become white vertices, sources black vertices, and a transmitting crossing becomes a white and a black vertex joined by an hourglass. The edge labels form the growth labeling, a proper labeling of the web with boundary word $w$. Here the rules are applied in rounds, scanning left to right; the Growth view shows the rounds, and the step slider replays them with the dangling labels.

What the page checks. For every sample it computes $\mathrm{prom}_\bullet(T)$ from the tableau and $\mathrm{trip}_\bullet(G)$ from the web independently and compares them, and it recovers $T$ from the web through the antiexcedances of $\mathrm{trip}_\bullet$ (the map $\mathcal T$ of [GPPSS]). The core code was also checked exhaustively for all $24\,024$ standard tableaux of shape $4\times 4$ and on random tableaux up to $c=60$.

Views. Disk: Tutte's harmonic embedding, boundary on the circle and each crossing at the average of its neighbours. Chords: each crossing is the meeting point of the straight chords of its two $\mathrm{trip}_2$ strands, which pass straight through every crossing and pairwise cross at most once, so the web becomes the chord diagram of the matching $\mathrm{prom}_2$; the slider hides strands of cyclic length below a threshold. Growth: the growth diagram, boundary on top. The mark at the top of the circle is the base face, between $b_n$ and $b_1$. Hover a boundary vertex, a matrix row or a tableau cell to see the $\mathrm{trip}_1$ (purple), $\mathrm{trip}_2$ (orange) and $\mathrm{trip}_3$ (green) strands from $b_i$; click to pin. Zoom the web with the + / − buttons, ⌘/Ctrl + scroll, or a trackpad pinch; drag to pan; double-click to reset.

Large $c$. In simulations the empirical permuton of $\mathrm{prom}_i$ concentrates on the diagonal, with the wrap-around at the corners: the fraction of $b$ with cyclic distance $|\mathrm{prom}_1(b)-b|>n/20$ is about $0.45,\,0.25,\,0.13,\,0.07$ for $n=100,400,1600,6400$, roughly $n^{-1/2}$, as for the noncrossing matching of a random Dyck path ($r=2$). Accordingly $\mathrm{inv}(\mathrm{prom}_1)/n^2\approx 0.133,\,0.080,\,0.043,\,0.024,\,0.012$ for $n=100,400,1600,6400,25600$, consistent with $\mathrm{inv}(\mathrm{prom}_i)\asymp n^{3/2}$: about $\sqrt n$ macroscopic jumps, each carrying order $n$ inversions. The large-scale structure sits in the few long $\mathrm{trip}_2$ chords, which the Chords view with a length threshold isolates.