The major index (maj) and its Sch\"utzenberger dual

Abstract

We construct the independent particle representation for the Semistandard Young Tableaux (SsYT) of skew shape $\lambda/\mu.$ The partition function of this particle system gives the generating function of the SsYT of skew shape $\lambda/\mu.$ Thus we obtain a bijective proof of the Stanley formula for the SsYT generating function. To do this we define for every SsYT $T$ its plinth, $\mathsf{p}\left( T\right) ,$ which is a SsYT of the same shape $\lambda/\mu.$ The set of plinths is finite. Our bijection associates to every SsYT $T$ a pair $\left( \mathsf{p}\left( T\right) ,Y\left( T-\mathsf{p}\left( T\right) \right) \right) ,$ where $Y\left( T-\mathsf{p}\left( T\right) \right) $ is the reading Young diagram of the SsYT $\left( T-\mathsf{p}\left( T\right) \right) $. \newline In particular, every Standard Young Tableau (SYT) $P$ has its plinth, $\mathsf{p}\left( P\right) $. The two statistics of SYT-s -- the volume $\left\vert \mathsf{p}\left( P\right) \right\vert $ and $\mathsf{maj}\left( P\right) $ -- are related via the Sch\"{u}tzenberger involution $Sch:$% \[ \left\vert \mathsf{p}\left( P\right) \right\vert =\mathsf{maj}\left( Sch\left( P\right) \right) . \]