$q$-Rationals and Finite Schubert Varieties

TL;DR

This paper explores $q$-analogues of rational numbers, connecting combinatorial interpretations with geometric structures like Schubert varieties, and introduces a novel approach using snake graphs to analyze these relationships.

Contribution

It introduces a new snake graph method to interpret $q$-rationals and links them to the geometry of Schubert varieties over finite fields.

Findings

$q$-rationals count sizes of certain varieties over finite fields.

Snake graphs provide a novel combinatorial interpretation.

Numerators of $q$-rationals relate to unions of Schubert cells.

Abstract

The classical $q$-analogue of the integers was recently generalized by Morier-Genoud and Ovsienko to give $q$-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 $q$-rationals count the sizes of certain varieties over finite fields, which are unions of open Schubert cells in some Grassmannian.