Special Year Seminar II

The ring of symmetric functions has a linear basis of Schur functions $s_{\lambda}$ indexed by partitions $\lambda = (\lambda_1 \geq \lambda_2 \geq \ldots \geq 0 )$. Littlewood-Richardson coefficients $c^{\nu}_{\lambda, \mu}$ 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 $s_{\lambda}s_{\mu}-s_{\rho}s_{\nu}$. To this end a lot of work has been done to study inequalities of such type. Lam--Postnikov--Pylyavskyy have shown Schur positivity of $s_{\lambda \cup \mu}s_{\lambda \cap \mu}-s_{\lambda}s_{\mu}$ 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 $s_{\lambda}s_{\mu}-s_{\rho}s_{\nu}$ which in particular implies positivity of $s_{\lambda \cup \mu}s_{\lambda \cap \mu}-s_{\lambda}s_{\mu}$. Based on recent work of Nguyen--Pylyavskyy on Temperley-Lieb immanants, we obtain several results. One of them is $c^{\nu}_{\lambda \cup \mu, \lambda \cap \mu}-c^{\nu}_{\lambda,\mu} \in \# P$. Another one states equality conditions of inequalities proved by Lam--Postnikov--Pylyavskyy.

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