The major index (maj) and its Schützenberger 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 $\left|\mathsf{p}\left(P\right)\right|$ and $\mathsf{maj}\left(P\right)$ -- are related via the Schützenberger involution $Sch:$% \[ \left\vert \mathsf{p}\left( P\right) \right\vert =\mathsf{maj}\left( Sch\left( P\right) \right) . \]

20 pages