Minimal Bar Tableaux

Details Minimal Bar Tableaux Minimal Bar Tableaux

2003-11-24

arxiv.org/abs/math/0311418v1

Mathematics

Combinatorics

12 pages, 7 figures

Scientific paper

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.

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