Schur Expansions Via Crystal Bases

Crystal bases were first introduced as a combinatorial tool to understand representations of quantum groups. They can also be understood from a purely combinatorial point of view (see for example the new book by Bump and Schilling on "Crystal bases: Representations and Combinatorics"). In particular, the character of a connected crystal in type A is a Schur function. Hence, knowing the crystal structure on a set that underlies a given symmetric function, will yield the Schur expansion of this symmetric function provided that the expansion is Schur-positive. As an example, Brendon Rhoades in his article “Ordered set partition statistics and the Delta conjecture” investigated the symmetric functions Val_{n,k}(x;0,q). These symmetric functions are Schur-positive (see page 19 of Rhoades' article). It would be interesting to find a crystal structure on the ordered multiset partitions that define these symmetric functions. Crystal bases are very well developed in the open source computer algebra system Sage. So it should be easy to experiment with these structures in Sage. References: * Brendon Rhoades, Ordered set partition statistics and the Delta conjecture arXiv:1605.04007 * Sage * Daniel Bump, Ben Salisbury, Anne Schilling Thematic Tutorial: Lie Methods and Related Combinatorics in Sage Non UBCUnreviewedAuthor affiliation: University of California DavisFacult