Linear time equivalence of Littlewood--Richardson coefficient symmetry maps

TL;DR

This paper demonstrates that various conjugation symmetry maps related to Littlewood-Richardson coefficients are linearly time reducible to each other, establishing their computational equivalence and answering a question by Pak and Vallejo.

Contribution

It explicitly constructs the Yamanouchi word for the conjugation symmetry map and proves the linear time reducibility among key symmetry maps and the Schutzenberger involution.

Findings

Conjugation symmetry map is linearly time reducible to Schutzenberger involution.

Various symmetry maps are linearly time equivalent.

Answers a question posed by Pak and Vallejo regarding computational complexity.

Abstract

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.