The srank conjecture on Schur’s Q-functions

William Y. C. Chen, Donna Q. J. Dou, Robert L. Tang and Arthur L. B. Yang

Vol. 244 (2010), No. 2, 211–234

DOI: 10.2140/pjm.2010.244.211

Abstract

We show that the shifted rank, or srank, of any partition λ with distinct parts equals the lowest degree of the terms appearing in the expansion of Schur’s Q_λ function in terms of power sum symmetric functions. This gives an affirmative answer to a conjecture of Clifford. As pointed out by Clifford, the notion of the srank can be naturally extended to a skew partition λ∕μ as the minimum number of bars among the corresponding skew bar tableaux. While the srank conjecture is not valid for skew partitions, we give an algorithm to compute the srank.

Keywords

srank, bar tableau, Schur’s Q-function, power sum symmetric function, strip tableau, skew bar tableau

Mathematical Subject Classification 2000

Primary: 05E05

Secondary: 20C25

Milestones

Received: 9 March 2009

Accepted: 14 March 2009

Published: 14 December 2009