Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A uniform action of the dihedral group $ Z_2\times D_3$ on Littlewood--Richardson coefficients

Olga Azenhas, Alessandro Conflitti, Ricardo Mamede

TL;DR

The paper establishes a faithful, uniform action of the group $\mathbb{Z}_2\times D_3$ on LR data (LR tableaux, their companions, KT W puzzles, and hives), organizing the symmetries into a manifest subgroup $H\simeq D_3$ and a complementary coset that encodes hidden commutativity and conjugation. Central to the approach is a suite of linear-time involutions (e, $\bullet$, $\blacklozenge$, $\spadesuit$, $\clubsuit$, and their composites) that realize LR transposers and commuters, all of which are linearly reducible to the Lusztig–Schützenberger involution. By transporting these involutions across models via the Tao puzzle bijection and Burge correspondence, the authors show that LR commuters and transposers coincide and are computable in linear time, with the migration/infusion framework in Purbhoo mosaics offering a unifying geometric intuition. The results give a complete, uniform account of the LR coefficient symmetries, translating puzzle dualities and triangle-hive relations into tableau operations and enabling efficient algorithms for symmetry transformations. The work thus ties together LR tableaux, hives, and puzzles under a single linear-time symmetry framework, clarifying the relationships among commutativity, conjugation, and dualities, and highlighting the central role of the Lusztig–Schützenberger involution in this uniform picture.

Abstract

We show that the dihedral group $ Z_2\times D_3$ of order twelve acts faithfully on the set LR, either consisting of Littlewood-Richardson tableaux, or their companion tableaux, or Knutson-Tao hives or Knutson-Tao-Woodward puzzles,via involutions which simultaneously conjugate or shuffle a Littlewood-Richardson triple of partitions. The action of $ Z_2\times D_3$ carries a linear time index two subgroup $H\simeq D_3$ action, where an involution which goes from $H$ into the other coset of H is difficult in the sense that it is not manifest neither exhibited by simple means. Pak and Vallejo have earlier made this observation with respect to the subgroup of index two in the symmetric group $ S_3$ consisting of cyclic permutations which H extends. The other half LR symmetries, not in the range of the H-action, are hidden and consist of commutativity and conjugation symmetries. Their exhibition is reduced to the action of a remaining generator of $ Z_2\times D_3$, which belongs to the other coset of H, and enables to reduce in linear time all known LR commuters and transposers to each other, and to the Luzstig- Schützenberger involution. A hive is specified by superimposing the companion tableau pair of an LR tableau, and its $Z_2\times D_3$-symmetries are exhibited via the corresponding LR companion tableau pair. The action of $ Z_2\times D_3$ on puzzles, naturally in bijection with Purbhoo mosaics, is consistent with the migration map on mosaics which translates to jeu de taquin slides or tableau-switching on LR tableaux. Their H-symmetries are reduced to simple procedures on a puzzle via label swapping together with simple reflections of an equilateral triangle, that is, puzzle dualities, and rotations on an equilateral triangle.

A uniform action of the dihedral group $ Z_2\times D_3$ on Littlewood--Richardson coefficients

TL;DR

The paper establishes a faithful, uniform action of the group on LR data (LR tableaux, their companions, KT W puzzles, and hives), organizing the symmetries into a manifest subgroup and a complementary coset that encodes hidden commutativity and conjugation. Central to the approach is a suite of linear-time involutions (e, , , , , and their composites) that realize LR transposers and commuters, all of which are linearly reducible to the Lusztig–Schützenberger involution. By transporting these involutions across models via the Tao puzzle bijection and Burge correspondence, the authors show that LR commuters and transposers coincide and are computable in linear time, with the migration/infusion framework in Purbhoo mosaics offering a unifying geometric intuition. The results give a complete, uniform account of the LR coefficient symmetries, translating puzzle dualities and triangle-hive relations into tableau operations and enabling efficient algorithms for symmetry transformations. The work thus ties together LR tableaux, hives, and puzzles under a single linear-time symmetry framework, clarifying the relationships among commutativity, conjugation, and dualities, and highlighting the central role of the Lusztig–Schützenberger involution in this uniform picture.

Abstract

We show that the dihedral group of order twelve acts faithfully on the set LR, either consisting of Littlewood-Richardson tableaux, or their companion tableaux, or Knutson-Tao hives or Knutson-Tao-Woodward puzzles,via involutions which simultaneously conjugate or shuffle a Littlewood-Richardson triple of partitions. The action of carries a linear time index two subgroup action, where an involution which goes from into the other coset of H is difficult in the sense that it is not manifest neither exhibited by simple means. Pak and Vallejo have earlier made this observation with respect to the subgroup of index two in the symmetric group consisting of cyclic permutations which H extends. The other half LR symmetries, not in the range of the H-action, are hidden and consist of commutativity and conjugation symmetries. Their exhibition is reduced to the action of a remaining generator of , which belongs to the other coset of H, and enables to reduce in linear time all known LR commuters and transposers to each other, and to the Luzstig- Schützenberger involution. A hive is specified by superimposing the companion tableau pair of an LR tableau, and its -symmetries are exhibited via the corresponding LR companion tableau pair. The action of on puzzles, naturally in bijection with Purbhoo mosaics, is consistent with the migration map on mosaics which translates to jeu de taquin slides or tableau-switching on LR tableaux. Their H-symmetries are reduced to simple procedures on a puzzle via label swapping together with simple reflections of an equilateral triangle, that is, puzzle dualities, and rotations on an equilateral triangle.
Paper Structure (44 sections, 28 theorems, 123 equations, 2 figures)

This paper contains 44 sections, 28 theorems, 123 equations, 2 figures.

Table of Contents

  1. Introduction
  2. LR coefficients as structure coefficients and symmetries
  3. The set $\mathcal{LR}$ and a representation of $\mathbb Z_2\times { D}_3$ in $\mathfrak{S}_{\mathcal{LR}}$
  4. The $H$-action and the action of a remaining generator of $\mathbb Z_2\times {D}_3$ on $\mathcal{LR}$
  5. Organization
  6. Preliminaries on tableau combinatorics
  7. Young (skew) diagrams and linear transformations
  8. Tableaux and Littlewood-Richardson tableaux
  9. The recording matrix of a tableau
  10. The linear involution rotation map $\bullet$ on SSYT's and LR tableaux
  11. Jeu de taquin and Burge correspondence
  12. Jeu de taquin
  13. Burge correspondence a variation of Robinson-Schensted-Knuth correspondence
  14. The recording tableau in Burge correspondence and LR tableaux
  15. Evacuation, reverse complementation, rotation and RSK
  16. ...and 29 more sections

Key Result

Proposition 2.1

The rotation map is a linear involution on $\rm LR(\mu,\nu,\lambda)\cup\rm\rm LR(\lambda,\nu^\bullet,\mu)$ that transforms the LR tableau $T$ into its dual $\bullet(T)$, the opposite LR tableau of shape $(\lambda^\vee/\mu)^\bullet$ and content $\nu^\bullet$, and vice versa. It exhibits the symmetry $c_{\mu,\nu,\lam

Figures (2)

  • Figure 5.1:
  • Figure :

Theorems & Definitions (77)

  • Example 2.1
  • Example 2.2
  • Proposition 2.1
  • Example 2.3
  • Definition 2.1
  • Proposition 2.2
  • Example 2.4
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • ...and 67 more