Classifying the near-equality of ribbon Schur functions

Foster Tom

Published 2018-10-01Version 1

We consider the problem of determining when the difference of two ribbon Schur functions is a single Schur function. We prove that this near-equality phenomenon occurs in sixteen infinite families and we conjecture that these are the only possible cases. Towards this converse, we prove that under certain additional assumptions the only instances of near-equality are among our sixteen families. In particular, we prove that our first ten families are a complete classification of all cases where the difference of two ribbon Schur functions is a single Schur function whose corresponding partition has at most two parts at least 2. We then provide a framework for interpreting the remaining six families and we explore some ideas toward resolving our conjecture in general. We also determine some necessary conditions for the difference of two ribbon Schur functions to be Schur-positive.