Equalities and Inequalities on Products of Schur Functions

The ring of symmetric functions has a linear basis of Schur functions sλ indexed by partitions λ=(λ1≥λ2≥…≥0). Littlewood-Richardson coefficients cνλ,μ are the structure constants of such a basis.

A function is Schur nonnegative if it is a linear combination with nonnegative coefficients of Schur functions. Some inequalities between Littlewood-Richardson coefficients are equivalent to Schur positivity of sλsμ−sρsν. To this end a lot of work has been done to study inequalities of such type. Lam--Postnikov--Pylyavskyy have shown Schur positivity of sλ∪μsλ∩μ−sλsμ as well as some other inequalities conjectured by Okounkov, by Fomin--Fulton--Li--Poon and by Lascoux-Leclerc-Thibon.

We suggest a necessary condition for Schur positivity of sλsμ−sρsν

which in particular implies positivity of sλ∪μsλ∩μ−sλsμ. Based on recent work of Nguyen--Pylyavskyy on Temperley-Lieb immanants, we obtain several results. One of them is cνλ∪μ,λ∩μ−cνλ,μ∈#P. Another one states equality conditions of inequalities proved by Lam--Postnikov--Pylyavskyy.

This is a joint work (in progress) with Igor Pak.