Preamble: Every n-letter word $w$ constructed from a n-letter alphabet is uniquely mapped by the RSK algorithm to a couple of tableaux P and Q, where P is a semi-standard Young tableau (SSYT) and Q is a standard Young tableau (SYT). P is then a SSYT with max entry n while Q is a SYT with n cells and both P and Q have the same shape. Evidently, the weight of $w$ is the same as the weight of $P$, and
$\sum_{|\lambda|=n} c_{\lambda,n} f_\lambda = n^n$
where $c_{\lambda,n}$ counts the SSYT of shape $\lambda$ with max entry $n$ .
I accidentatly stumbled on the following symmetries:
$S(P)°S(Q) = n+1- ( P ° Q )$
$S(P)° Q = n+1- ( P °S(Q))$
$ P °S(Q) = n+1- (S(P)° Q )$
$ P ° Q = n+1- (S(P)°S(Q))$
where $S(T)$ is the Schutzenberger dual tableau to T,and $P°Q$ is shorthand for the word generated by the (reverse) RSK algorithm on tableaux $P$ and $Q$. Also, an integer $n+1$ minus a word $w$ signifies the 'complement' word $( n+1-w_1, n+1-w_2, ..)$
Actual question : are any other tableau symmetries known, corresponding to simple reversion of the word $w$, or to the 'complement', or to the rotations of the word $w$?
Actually,
this looks like bi-pyramidal symmetry $D_{n h}$ on the set of all couples $P,Q$. Is that even correct?
