The srank Conjecture on Schur's $Q$-Functions

TL;DR

This paper proves that the shifted rank of partitions with distinct parts equals the lowest degree in the Schur's Q-function expansion, confirming Clifford's conjecture for partitions and providing an algorithm for skew partitions.

Contribution

It establishes the equality of shifted rank and minimal degree in Q-function expansion for partitions and extends the concept to skew partitions with an algorithm.

Findings

Confirmed the srank conjecture for partitions with distinct parts.

Provided an algorithm to compute the srank for skew partitions.

Clarified the limitations of the srank conjecture for skew partitions.

Abstract

We show that the shifted rank, or srank, of any partition $\lambda$ with distinct parts equals the lowest degree of the terms appearing in the expansion of Schur's $Q_{\lambda}$ 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 $\lambda/\mu$ 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.