KnuthClassRepresentative from Permutation
The row reading word of σ's RSK insertion tableau — the canonical word of its Knuth (plactic) class.
Two permutations are Knuth-equivalent exactly when they share an insertion tableau (Schensted), so reading that tableau back out — bottom row to top, left to right — picks one fixed representative per class. Idempotent: the representative's own insertion tableau is the same P, so applying this again changes nothing.
Called as CombinatorialMap(x, "KnuthClassRepresentative"). The head's page has its examples.