Abstract
We study the relationship between rational slope Dyck paths and invariant subsets in Z, extending the work of the first two authors in the relatively prime case. We also find a bijection between (dn, dm)–Dyck paths and d-tuples of (n, m)-Dyck paths endowed with certain gluing data. These are first steps towards understanding the relationship between the rational slope Catalan combinatorics in non relatively prime case and the geometry of affine Springer fibers and representation theory.
Highlights
Catalan numbers, in one of their incarnations, count the number of Dyck paths, that is, the lattice paths in a square which never cross the diagonal
For coprime m and n there are a number of interesting maps involving Yn,m: (a) J
We introduce a certain equivalence relation ∼ on MN,M
Summary
In one of their incarnations, count the number of Dyck paths, that is, the lattice paths in a square which never cross the diagonal. One ranks the steps of the boundary path of D as follows. Note that for relatively prime (n, m) all the ranks of the steps of a diagram D ∈ Yn,m are distinct.
Sign-in/Register to view highlights & summary 
Published Version (
Free)
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call