The [Robinson-Schensted correspondence](https://en.wikipedia.org/wiki/Robinson%E2%80%93Schensted%E2%80%93Knuth_algorithm) is a bijection between elements of the symmetric group and ordered pairs of standard tableaux of the same shape.


Some simple operations on tableaux correspond to simple operations on the group: switching the tableaux corresponds to inverse on the group.



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> What about taking the transpose of the tableaux? Does that correspond to something easily described on permutations?
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In order to rule out easy guesses, let me describe this on $S\_3$:


* the identity switches with the involution 321.
* the transpositions 213 and 132 switch.
* the 3-cycles 312 and 231 switch.


In general, this operation preserves being order $\leq 2$ (since this is equivalent to the P- and Q-symbols being the same).



What bijection on permutations corresponds under RS to transpose?
When you conjugate diagrams and apply RSK or other Young tableau bijections, the answer is typically bad, with some rare exceptions. The right way to think of RSK is to think of row length being continuous while columns still integer (see e.g. my "Geometric proof of the hook-length formula" paper and refs therein). 


An exception: there is a "hidden symmetry" for LR-coefficients when you conjugate all three diagrams - see Hanlon-Sundaram paper (1992). Once you know this bijection, there are natural connections to RSK as described in the long Pak-Vallejo paper and a followup by Azenhas-Conflitti-Mamede (search the web for a paper and a ppt presentation). Together, these do give a complete description of your involution, but making sense of it might require quite a bit of work. 


