On an index two subgroup of puzzle and Littlewood-Richardson tableau Z2 x S3-symmetries

Abstract

We consider an action of the dihedral group Z2 × S3 on Littlewood- Richardson tableaux which carries a linear time action of a subgroup of index two. This index two subgroup action on Knutson-Tao-Woodward puzzles is the group generated by the puzzle mirror reflections with label swapping. One shows that, as happens in puzzles, half of the twelve symmetries of Littlewood-Richardson coefficients may also be exhibited on Littlewood-Richardson tableaux by surprisingly easy maps. The other hidden half symmetries are given by a remaining generator which enables to reduce those symmetries to the Sch¨utzenberger involution. Purbhoo mosaics are used to map the action of the subgroup of index two on Littlewood- Richardson tableaux into the group generated by the puzzle mirror reflections with label swapping. After Pak and Vallejo one knows that Berenstein-Zelevinsky triangles, Knutson-Tao hives and Littlewood-Richardson tableaux may be put in correspondence by linear algebraic maps. We conclude that, regarding the symmetries, the behaviour of the various combinatorial models for Littlewood-Richardson coefficients is similar, and the bijections exhibiting them are in a certain sense unique.