$q$-Rationals and Finite Schubert Varieties

Abstract

The classical q-analogue of the integers was recently generalized by Morier-Genoud and Ovsienko to give qanalogues 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.

$q$-Rational and $q$-Real Binomial Coefficients

“…the number of boxes) underneath the path p, then R(q) = p∈L(Gα) q |p| . This is an equivalent re-statement of Theorem 4 from [MGO20], which was stated in the present form (in terms of the snake graph language) in [Cla20] and [Ove21]. Furthermore, the denominator S(q) has the same interpretation, but for the smaller snake graph obtained from G α by removing the initial vertical column of boxes.…”

“…This interpretation, in terms of lattice paths and skew Young diagrams, was used in [Ove21] to give another combinatorial meaning of the polynomials R(q) as counting the sizes of certain varieties over the finite field with q elements. In particular, q |µ| R(q) is the number of F q -points in a union of Schubert cells in some Grassmannian, where the union is over the Schubert cells indexed by partitions ν with µ ≤ ν ≤ λ.…”

Quantum continuants, quantum rotundus and triangulations of annuli

“…The subject has led to further developments in various directions. Notably there are established links with knots invariants [22], [18], the modular group and the Picard group [20], [33], combinatorics of posets [23], [30], [31], Markov numbers and Markov-Hurwitz approximation theory [9], [17], [19], [21], geometry of Grassmannians [32], triangulated categories [1].…”