Linear time equivalence of Littlewood--Richardson coefficient symmetry maps

Benkart, Sottile, and Stroomer have completely characterized by Knuth and dual Knuth equivalence a bijective proof of the conjugation symmetry of the Littlewood-Richardson coefficients. Tableau-switching provides an algorithm to produce such a bijective proof. Fulton has shown that the White and the Hanlon-Sundaram maps are versions of that bijection. In this paper one exhibits explicitly the Yamanouchi word produced by that conjugation symmetry map which on its turn leads to a new and very natural version of the same map already considered independently. A consequence of this latter construction is that using notions of Relative Computational Complexity we are allowed to show that this conjugation symmetry map is linear time reducible to the Schutzenberger involution and reciprocally. Thus the Benkart-Sottile-Stroomer conjugation symmetry map with the two mentioned versions, the three versions of the commutative symmetry map, and Schutzenberger involution, are linear time reducible to each other. This answers a question posed by Pak and Vallejo.