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Promotion permutations and the Robinson--Schensted correspondence

Stephan Pfannerer, Joshua P. Swanson

TL;DR

The paper connects promotion permutations of rectangular standard Young tableaux to the Robinson–Schensted correspondence by stacking two tableaux and examining the r-th northeast promotion; it proves $\mathsf{prom}_r^{\mathrm{NE}}(\mathrm{stack}(P,Q))=\mathrm{RS}^{-1}(Q^{\top},P^{\top})$ and shows that the full northeast/southwest promotion data recovers Viennot's shadow-line construction. It further characterizes the image of the promotion-tuple map in terms of fixed-point-free involutions with prescribed crossing and nesting numbers, linking promotion combinatorics to geometric shadow lines and RS statistics. The main theorem provides a precise description of how the promotion matrix decomposes into NE, SE, SW, NW blocks via skeleta, yielding a unified framework that encompasses the RS correspondence, Viennot’s construction, and crossing/nesting phenomena in the rectangular setting. The results give a geometric interpretation of promotion symmetries and advance understanding of how promotion permutations encode RS data in a structured, quadrant-wise manner. This work thus bridges promotion theory, RS combinatorics, and Viennot’s shadow-line geometry with potential implications for related growth-diagram constructions and representation-theoretic interpretations.

Abstract

Promotion permutations have recently been associated to each rectangular standard Young tableau by Gaetz--Pechenik--Pfannerer--Striker--Swanson. Here we relate promotion permutations to the Robinson--Schensted (RS) correspondence. More precisely, we show that taking a pair of standard Young tableaux of the same rectangular shape, stacking them, and computing the middle promotion permutation yields the RS permutation of the pair up to simple twists. Moreover, the full list of promotion permutations in this special case encodes Viennot's geometric shadow line construction. As a consequence, we characterize a subset of the collection of possible promotion permutations in terms of crossing and nesting numbers.

Promotion permutations and the Robinson--Schensted correspondence

TL;DR

The paper connects promotion permutations of rectangular standard Young tableaux to the Robinson–Schensted correspondence by stacking two tableaux and examining the r-th northeast promotion; it proves and shows that the full northeast/southwest promotion data recovers Viennot's shadow-line construction. It further characterizes the image of the promotion-tuple map in terms of fixed-point-free involutions with prescribed crossing and nesting numbers, linking promotion combinatorics to geometric shadow lines and RS statistics. The main theorem provides a precise description of how the promotion matrix decomposes into NE, SE, SW, NW blocks via skeleta, yielding a unified framework that encompasses the RS correspondence, Viennot’s construction, and crossing/nesting phenomena in the rectangular setting. The results give a geometric interpretation of promotion symmetries and advance understanding of how promotion permutations encode RS data in a structured, quadrant-wise manner. This work thus bridges promotion theory, RS combinatorics, and Viennot’s shadow-line geometry with potential implications for related growth-diagram constructions and representation-theoretic interpretations.

Abstract

Promotion permutations have recently been associated to each rectangular standard Young tableau by Gaetz--Pechenik--Pfannerer--Striker--Swanson. Here we relate promotion permutations to the Robinson--Schensted (RS) correspondence. More precisely, we show that taking a pair of standard Young tableaux of the same rectangular shape, stacking them, and computing the middle promotion permutation yields the RS permutation of the pair up to simple twists. Moreover, the full list of promotion permutations in this special case encodes Viennot's geometric shadow line construction. As a consequence, we characterize a subset of the collection of possible promotion permutations in terms of crossing and nesting numbers.

Paper Structure

This paper contains 13 sections, 10 theorems, 34 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Background
  4. Promotion and evacuation
  5. Promotion permutations
  6. Promotion permutations
  7. Local rules, growth diagrams, and decorations
  8. Existing work and Fomin's local rules
  9. Crossing and nesting numbers
  10. Viennot's construction
  11. Main theorem and proof
  12. Main theorem statement
  13. Main theorem proof

Key Result

Theorem 1.1

If $P, Q$ are rectangular standard Young tableaux with $r$ rows, then

Figures (3)

  • Figure 1: Example of stacking tableaux. Here we add $3 \times 4 = 12$ to the entries in $Q$ and place them below $P$.
  • Figure 2: The perfect matching of $\mathop{\mathrm{\mathsf{prom}}}\nolimits_3(\mathop{\mathrm{stack}}\nolimits(P, Q))$ for $P, Q \in \mathop{\mathrm{SYT}}\nolimits(3 \times 4)$ from \ref{['ex:prom-RS']}. Observe that the maximum number of mutually crossing strands is $3$ (e.g., $\{1, 19\}, \{2, 22\}, \{4, 24\}$) and the maximum number of nested strands is $4$ (e.g., $\{1, 19\}, \{3, 15\}, \{5, 14\}, \{6, 13\})$, in agreement with \ref{['cor:prom-RS']}.
  • Figure 3: (Left) A collection $\mathcal{P}$ depicted as filled points, together with the shadows cast with respect to light shining $\nearrow$. The shadow lines partition $\mathcal{P}$ into four pieces. The skeleton $\mathcal{S}(\mathcal{P}, \nearrow)$ is depicted as unfilled circles. (Right) The union of the shadow lines for all skeleta of $\mathcal{P}$ drawn simultaneously. Each $i$-skeleton $\mathcal{S}^i$ for $0 \leq i \leq 3$ is in a different color and is labeled at the right and top with $i+1$. Reading these labels gives lattice words $w_P = 1213141$, $w_Q = 1211341$. These words encode successive row indexes of the indicated RS tableaux $(P, Q)$, which corresponds under RS to the permutation $\rho = [6, 2, 4, 5, 3, 1, 7]$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Example 1.2
  • Corollary 1.3
  • Remark 1.4
  • Example 2.1
  • Definition 2.2
  • Example 2.3
  • Theorem 2.4: fluctuating-paper
  • Remark 2.5
  • Example 2.6
  • ...and 31 more