21/09/2015, 11:40 — 12:30 — Room P3.10, Mathematics Building
Olga Azenhas, Universidade de Coimbra
Schur positivity and ribbon shapes with interval support
The ring of symmetric functions has the basis of Schur functions $s_\lambda$, indexed by partitions $\lambda$. A symmetric function is said to be Schur positive if when expanded as a linear combination of Schur functions all the coefficient are non negative integers. Skew Schur functions and the product of two Schur functions are examples of Schur positive functions where the coefficients in the Schur expansion are the Litlewood-Richardson coefficients.
For any skew shape $A$, the support of $A$ (or $ s_A$) is defined to be those partitions $\lambda$ such that the Schur function $s_\lambda$ appears with positive coefficient in the Schur expansion of $s_A$ (assuming infinitely many variables). Let $\operatorname{rows}(A)$ denote the partition formed by sorting the row lengths of $A$ into weakly decreasing order, and define $\operatorname{cols}(A)$ similarly. It is well known that the support of $ A$, considered as a subposet of the dominance lattice, has a top element $\operatorname{cols}(A)^t$ (the conjugate of $\operatorname{cols}(A)$) and a bottom element $\operatorname{rows}(A)$.
We seek to understand how the support of $A$ is governed by the shape of $A$, in particular, when the whole interval $ [\operatorname{rows}(A), \operatorname{cols}(A)^t]$, in the dominance lattice, is attained. We focus our attention on ribbon and disjoint union of ribbon shapes. This is a joint work with Ricardo Mamede.
See also
lisbon-sept-2015.pdf
