14 research outputs found
A minimaj-preserving crystal on ordered multiset partitions
We provide a crystal structure on the set of ordered multiset partitions,
which recently arose in the pursuit of the Delta Conjecture. This conjecture
was stated by Haglund, Remmel and Wilson as a generalization of the Shuffle
Conjecture. Various statistics on ordered multiset partitions arise in the
combinatorial analysis of the Delta Conjecture, one of them being the minimaj
statistic, which is a variant of the major index statistic on words. Our
crystal has the property that the minimaj statistic is constant on connected
components of the crystal. In particular, this yields another proof of the
Schur positivity of the graded Frobenius series of the generalization
due to Haglund, Rhoades and Shimozono of the coinvariant algebra . The
crystal structure also enables us to demonstrate the equidistributivity of the
minimaj statistic with the major index statistic on ordered multiset
partitions.Comment: 17 pages; v2 contains minor changes suggested by referee, references
update
Affine structures and a tableau model for E_6 crystals
We provide the unique affine crystal structure for type E_6^{(1)}
Kirillov-Reshetikhin crystals corresponding to the multiples of fundamental
weights s Lambda_1, s Lambda_2, and s Lambda_6 for all s \geq 1 (in Bourbaki's
labeling of the Dynkin nodes, where 2 is the adjoint node). Our methods
introduce a generalized tableaux model for classical highest weight crystals of
type E and use the order three automorphism of the affine E_6^{(1)} Dynkin
diagram. In addition, we provide a conjecture for the affine crystal structure
of type E_7^{(1)} Kirillov-Reshetikhin crystals corresponding to the adjoint
node.Comment: 28 page