Minimal Bar Tableaux

Abstract

Motivated by Stanley's results in \cite{St02}, we generalize the rank of a partition $\lambda$ to the rank of a shifted partition $S(\lambda )$. We show that the number of bars required in a minimal bar tableau of $S(\lambda )$ is max$(o,e+(\ell (\lambda )\mathrm{mod}2))$, where $o$ and $e$ are the number of odd and even rows of $\lambda$. As a consequence we show that the irreducible projective characters of $S_n$ vanish on certain conjugacy classes. Another corollary is a lower bound on the degree of the terms in the expansion of Schur's $Q_{\lambda }$ symmetric functions in terms of the power sum symmetric functions.

12 pages, 7 figures