Bounds for nonzero Littlewood-Richardson coefficients (Müge Taskin, Boğaziçi University, Turkey)

As  $\lambda$ runs through all integer partitions, the set of   Schur functions $\{s_{\lambda}\}_\lambda$ forms a basis in the ring of symmetric functions. Hence the rule $$s_{\lambda}s_{\mu}=\sum c_{\lambda,\mu}^{\gamma} s_{\gamma}$$ makes sense and the coefficients $c_{\lambda,\mu}^{\gamma}$ are called \textit{Littlewood-Richardson (LR) coefficients}. The calculations of Littlewood-Richardson coefficients has been an important problem from the first time they were introduced, due to their important role in representation theory of symmetric groups and enumerative geometry.

In this talk we will explain some of the main features of these coefficients and provide a summary of the characterizations given by Littlewood and Richardson (1934), Berenstein- Zelevinsky ()1988) and Knutson-Tao (1999). Then we will explain our approach to a seemingly easier problem, that is, the determination of  triples $(\lambda,\mu,\gamma)$  of partitions for which $c_{\lambda,\mu}^{\gamma}$ is non zero. Our method describes some upper and lower bounds for triples $(\lambda,\mu,\gamma)$ with nonzero  $c_{\lambda,\mu}^{\gamma}$, by using  Young diagram combinatorics and especially, the indispensable Dominance order. This is joint work with R. Bedii Gümüş and supported by Tübitak/1001/115F156.