The srank Conjecture on Schur's Q-Functions
William Y.C. Chen, Donna Q. J. Dou, Robert L. Tang, and Arthur L. B. Yang
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 Qλ 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.
AMS Classification: 05E05, 20C25
Keywords: srank, bar tableau, Schur's Q-function, power sum symmetric function, strip tableau, skew bar tableau