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Local algorithms for coplactic switching and evacuation of Young tableaux

Kelsey M. Brown, Derek Moran

TL;DR

This work introduces local, shape-preserving algorithms for evacuation shuffling on pairs of skew Young tableaux, defining tableau coswitching as a coplactic analogue of classical tableau switching. It presents two concrete local realizations of $\mathrm{esh}$—a Pieri/jeux-de-taquin–style hopping method and a crystal-operator method—that compute $\mathrm{esh}$ on pairs $(X,T)$ with $X$ arbitrary and $T$ LR, and proves their outputs concur with coswitching. The authors extend these constructions to type B (shifted tableaux) and connect the combinatorics to geometric monodromy on smooth covers of the real moduli space $\overline{M}_{0,r}$, deriving computational and theoretical consequences including complexity improvements and lambda-dominant transition data. They also analyze inverses, variations, and the lambda-dominant structure of transition data, and discuss implications for real Schubert calculus, fixed points, and related combinatorial-geometric phenomena. Overall, the paper provides efficient, local, coplactic algorithms that illuminate the interaction between tableau combinatorics, crystal theory, and the geometry of moduli spaces, with broad applicability to both classical and shifted tableau settings.

Abstract

Tableau switching is a well studied bijection on pairs of skew Young tableaux which swaps their relative positions. This is achieved by successively sliding the entries of the inner tableaux through the outer one via jeu de taquin (JDT) slides. Tableau coswitching is a similar but coplactic operation, meaning it commutes with any sequence of JDT slides. Coswitching is defined by first performing JDT rectification on the union of the tableaux, switching the resulting pair, then unrectifying the union. This definition requires us to perform large scale modifications to the skew shapes during the rectification and unrectification steps, which is both computationally taxing and obscures the effect of coswitching on the pair of skew tableaux. In previous work, Gillespie and Levinson define an algorithm which computes coswitching as a sequence of local moves, which do not alter the skew shapes of the tableaux, when one of the tableaux is a single box. In this paper, we extend the results of Gillespie and Levinson (and Gillespie, Levinson, and Purbhoo for type B) to the case the two skew tableaux are of arbitrary size. We also describe multiple bijections on tableaux, each descending to the evacuation shuffling (esh) operation on pairs of dual equivalence classes. We show how special cases of our local algorithm compute the Schützenberger involution on skew tableaux, as well as the wall-crossing and monodromy of certain covering spaces of the moduli space $\overline{M}_{0, r}(\mathbb{R})$.

Local algorithms for coplactic switching and evacuation of Young tableaux

TL;DR

This work introduces local, shape-preserving algorithms for evacuation shuffling on pairs of skew Young tableaux, defining tableau coswitching as a coplactic analogue of classical tableau switching. It presents two concrete local realizations of —a Pieri/jeux-de-taquin–style hopping method and a crystal-operator method—that compute on pairs with arbitrary and LR, and proves their outputs concur with coswitching. The authors extend these constructions to type B (shifted tableaux) and connect the combinatorics to geometric monodromy on smooth covers of the real moduli space , deriving computational and theoretical consequences including complexity improvements and lambda-dominant transition data. They also analyze inverses, variations, and the lambda-dominant structure of transition data, and discuss implications for real Schubert calculus, fixed points, and related combinatorial-geometric phenomena. Overall, the paper provides efficient, local, coplactic algorithms that illuminate the interaction between tableau combinatorics, crystal theory, and the geometry of moduli spaces, with broad applicability to both classical and shifted tableau settings.

Abstract

Tableau switching is a well studied bijection on pairs of skew Young tableaux which swaps their relative positions. This is achieved by successively sliding the entries of the inner tableaux through the outer one via jeu de taquin (JDT) slides. Tableau coswitching is a similar but coplactic operation, meaning it commutes with any sequence of JDT slides. Coswitching is defined by first performing JDT rectification on the union of the tableaux, switching the resulting pair, then unrectifying the union. This definition requires us to perform large scale modifications to the skew shapes during the rectification and unrectification steps, which is both computationally taxing and obscures the effect of coswitching on the pair of skew tableaux. In previous work, Gillespie and Levinson define an algorithm which computes coswitching as a sequence of local moves, which do not alter the skew shapes of the tableaux, when one of the tableaux is a single box. In this paper, we extend the results of Gillespie and Levinson (and Gillespie, Levinson, and Purbhoo for type B) to the case the two skew tableaux are of arbitrary size. We also describe multiple bijections on tableaux, each descending to the evacuation shuffling (esh) operation on pairs of dual equivalence classes. We show how special cases of our local algorithm compute the Schützenberger involution on skew tableaux, as well as the wall-crossing and monodromy of certain covering spaces of the moduli space .

Paper Structure

This paper contains 21 sections, 21 theorems, 18 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Acknowledgements
  4. Notation and background
  5. Partitions and tableaux
  6. Jeu de taquin and equivalence relations
  7. Littlewood-Richardson tableaux
  8. Switching operations on tableaux and dual equivalence classes
  9. Connection to smooth covers of M0,n-bar
  10. Coplactic operators
  11. Main results
  12. Local evacuation shuffling
  13. Related Results
  14. Variations of the algorithms
  15. Reversing the algorithms
  16. ...and 6 more sections

Key Result

Theorem 1.1

Our algorithms compute coswitching and partial evacuation shuffling on pairs $(X, T)$ with $X$ arbitrary and $T$ LR, and the Schützenberger involution on arbitrary skew tableaux, in terms of local moves.

Figures (2)

  • Figure 1: An example of $\mathrm{esh}$ as a lift of the path $s_{2,3}$ in $\overline{M}_{0,4}$.
  • Figure 2: An example of $\mathrm{pesh}$ as a lift of the path $s_{2,7}$ in $\overline{M}_{0,8}$. Note that in the output, the blue entries $1,\ldots, 5$ of the standard Young tableau, correspond to the boxes $\text{\scriptsize \yng(1)}_5,\ldots,\text{\scriptsize \yng(1)}_1$ (in that order).

Theorems & Definitions (87)

  • Theorem 1.1: Theorems \ref{['thm: hop=esh']} and \ref{['thm: hop=crystal']}
  • Definition 2.1
  • Proposition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • ...and 77 more