$q$-Rationals and Finite Schubert Varieties
Combinatorics
2021-11-16 v1
Abstract
The classical -analogue of the integers was recently generalized by Morier-Genoud and Ovsienko to give -analogues of rational numbers. Some combinatorial interpretations are already known, namely as the rank generating functions for certain partially ordered sets. We review some of these interpretations, and additionally give a slightly novel approach in terms of planar graphs called snake graphs. Using the snake graph approach, we show that the numerators of -rationals count the sizes of certain varieties over finite fields, which are unions of open Schubert cells in some Grassmannian.
Cite
@article{arxiv.2111.07912,
title = {$q$-Rationals and Finite Schubert Varieties},
author = {Nicholas Ovenhouse},
journal= {arXiv preprint arXiv:2111.07912},
year = {2021}
}