Let $LR(\mu/\lambda;\nu)$ be the set of Littlewood-Richardson tableaux of shape $\mu/\lambda$ and weight $\nu$. Then there is a canonical bijection between $LR(\mu/\lambda;\nu)$ and $LR(\mu/\nu;\lambda)$, presented in a paper by Pak and Vallejo ("Fundamental Symmetry map"), and in a paper by Danilov and Koshevoi ("Commutor"). "Commutor"), and in a paper by Henriques and Kamnitz.
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In the paper by Pak and Vallejo actually two "Fundamental Symmetry maps" are presented. Danilov and Koshevoi show that they coincide, and that they coincide with their "commutor".commutor", and with the map defined by Henriques and Kamnitzer.
Igor Pak and Ernesto Vallejo.
Reductions of Young tableau bijections
SIAM J. Discrete Math. 24 (2010), no. 1, 113--145.
V.I. Danilov and G.A. Koshevoi
The Robinson-Schensted-Knuth correspondence and the bijections of commutativity and associativity.
2008 Izv. Math. 72 689
A. Henriques and J. Kamnitzer
The octahedron recurrence and $gl_n$-crystals
Adv. Math. 206:1 (2006), 211-249