On the Schur Positivity of Differences of Products of Schur Functions

The Schur functions are a basis for the ring of symmetric functions indexed by partitions of nonnegative integers. A symmetric function f is called Schur positive if when expressed as a linear combination of Schur functions

each coefficient cλ is nonnegative. We wish to investigate expressions of the form

where λ partitions n and μ partitions n-1 and the complements λc,μc are taken over a sufficiently large m×m square. We give a necessary condition that if (1) is Schur positive, then μ is contained in λ. Furthermore, we show how conjugating partitions preserve Schur positivity. Lastly, we incorporate the Littlewood Richardson rule to show that particular classes of λ of μ are never Schur positive.

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