Symmetric functions, shifted tableaux, and a class of distinct Schur Q-functions - Salois, Kyle, author; Gillespie, Maria, advisor; Cavalieri, Renzo, committee member; Hulpke, Alexander, committee member; Cooley, Daniel, committee member (Colorado State University. Libraries, 2022)
The Schur Q-functions form a basis of the algebra Ω of symmetric functions generated by the odd-degree power sum basis pd, and have ramifications in the projective representations of the symmetric group. So, as with ordinary Schur functions, it is relevant to consider the equality of skew Schur Q-functions Qλ/μ. This has been studied in 2008 by Barekat and van Willigenburg in the case when the shifted skew shape λ/μ is a ribbon. Building on this premise, we examine the case of near-ribbon shapes, formed by adding one box to a ribbon skew shape. We particularly consider frayed ribbons, that is, the near-ribbons whose shifted skew shape is not an ordinary skew shape. We conjecture with evidence that all Schur Q-functions for frayed ribbon shapes are distinct up to antipodal reflection. We prove this conjecture for several infinite families of frayed ribbons, using a new approach via the "lattice walks'' version of the shifted Littlewood-Richardson rule, discovered in 2018 by Gillespie, Levinson, and Purbhoo.
Keywords: shifted tableaux; Schur Q-functions; symmetric functions