Inequality of a class of near-ribbon skew Schur \(Q\)-functions (Q6161654)

The Schur \(Q\)-functions \(Q_\lambda(x_1, x_2,\dots)\), originally defined by \textit{I. Schur} [J. Reine Angew. Math. 139, 155--250 (1911; JFM 42.0154.02)], are analogues of the classical Schur functions for shifted partitions \(\lambda\) and are themselves symmetric functions. The Schur \(Q\)-functions naturally arise in various fields such as the projective representation theory of the symmetric group, the crystal base theory of the quantum queer Lie superalgebra, and the intersection theory of Schubert varieties in the odd orthogonal Grassmannian. The problem of determining when two skew Schur \(Q\)-functions are equal is still largely open. It has been studied in the case of ribbon shapes in 2008 by \textit{F. Barekat} and \textit{S. van Willigenburg} [Electron. J. Comb. 16, No. 1, Research Paper R110, 28 p. (2009; Zbl 1226.05256)]. The authors of this paper approach the problem for near-ribbon shapes, formed by adding one box to a ribbon skew shape. They introduce the notion of frayed ribbons, that is, the near-ribbons whose shifted skew shape is not an ordinary skew shape. In this paper, it is conjectured that all Schur \(Q\)-functions of frayed ribbon shape are distinct up to antipodal reflection. Using a new approach, supported by the ``lattice walks'' interpretation of the Littlewood-Richardson coefficients given in 2018 by \textit{M. Gillespie} et al. [Discrete Comput. Geom. 69, No. 4, 981--1039 (2023; Zbl 1512.05401)], the authors prove the conjecture for several infinite families of frayed ribbons.