The Plethysm sλ[sμ] at Hook and Near-Hook Shapes

We completely characterize the appearance of Schur functions corresponding to partitions of the form ν = (1a, b) (hook shapes) in the Schur function expansion of the plethysm of two Schur functions, sλ[sμ] = ∑ ν aλ,μ,νsν . Specifically, we show that no Schur functions corresponding to hook shapes occur unless λ and μ are both hook shapes and give a new proof of a result of Carbonara, Remmel and Yang that a single hook shape occurs in the expansion of the plethysm s(1c,d)[s(1a,b)]. We also consider the problem of adding a row or column so that ν is of the form (1a, b, c) or (1a, 2b, c). This proves considerably more difficult than the hook case and we discuss these difficulties while deriving explicit formulas for a special case.