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

TL;DR

This paper demonstrates a faithful action of the dihedral group Z_2×D_3 on Littlewood-Richardson objects, revealing hidden symmetries and providing linear-time methods to relate various LR transformations and involutions.

Contribution

It introduces a unified group action framework on LR tableaux, hives, and puzzles, uncovering hidden symmetries and simplifying known LR transformations.

Findings

Z_2×D_3 acts faithfully on LR objects.

Linear-time reduction of LR symmetries and involutions.

Exhibition of hidden symmetries via group actions.

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- Sch\"utzenberger 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.