arXiv subjects
Publications and source records attributed to Sophie Morier-Genoud.
We introduce a $q$-deformation of the Pythagoras equation $a^2 + b^2 = c^2$, which is a polynomial version of it different from the standard one. We construct a polynomial analogue, or ``$q$-analogue'', of every primitive Pythagorean triple. We also construct such analogue for a larger class of Pythagorean triples called standard. Our approach is based on the notion of $q$-deformed rational numbers and the modular group $\mathrm{PSL}(2,\mathbb{Z})$.
We consider a natural $q$-deformation of the classical Markov numbers. This $q$-deformation is closely related to $q$-deformed rational numbers recently introduced by two of us. Both notions, those of $q$-rationals and $q$-Markov numbers, are based on invariance with respect to the action of the modular group $mathrm{PSL}(2,\mathbb{Z})$. We prove that every Markov number has a unique $q$-deformation, which is a monic unimodal palindromic Laurent polynomial with positive integer coefficients. The $q$-Markov numbers can be calculated in terms of the traces of $q$-deformed Cohn matrices, and we show that $q$-Markov numbers are independent of the choice of such matrices. We construct a combinatorial model counting perfect matchings of snake graphs with weighted edges.
The concept of $q$-deformation, or ``$q$-analogue'' arises in many areas of mathematics. In algebra and representation theory, it is the origin of quantum groups; $q$-deformations are important for knot invariants, combinatorial enumeration, discrete geometry, analysis, and many other parts of mathematics. In mathematical physics, $q$-deformations are often understood as ``quantizations''. The recently introduced notion of a $q$-deformed real number is based on the geometric idea of invariance by a modular group action. The goal of this lecture is to explain what is a $q$-rational and a $q$-irrational, demonstrate beautiful properties of these objects, and describe their relations to many different areas. We also tried to describe some applications of $q$-numbers.
We give enumerative interpretations of the polynomials arising as numerators and denominators of the $q$-deformed rational numbers introduced by Morier-Genoud and Ovsienko. The considered polynomials are quantum analogues of the classical continuants and of their cyclically invariant versions called rotundi. The combinatorial models involve triangulations of polygons and annuli. We prove that the quantum continuants are the coarea-generating functions of paths in a triangulated polygon and that the quantum rotundi are the (co)area-generating functions of closed loops on a triangulated annulus.
We establish a link between the new theory of $q$-deformed rational numbers and the classical Burau representation of the braid group $\mathrm{B}_3$. We apply this link to the open problem of classification of faithful complex specializations of this representation. As a result we provide an answer to this problem in terms of the singular set of the $q$-rationals and prove the faithfulness of the Burau representation specialized at complex $t\in \mathbb{C}^*$ outside the annulus $3-2\sqrt2 \leq |t| \leq 3+2\sqrt2.$
We study analytic properties of ``$q$-deformed real numbers'', a notion recently introduced by two of us. A $q$-deformed positive real number is a power series with integer coefficients in one formal variable~$q$. We study the radius of convergence of these power series assuming that $q$ is a complex variable. Our main conjecture, which can be viewed as a $q$-analogue of Hurwitz's Irrational Number Theorem, claims that the $q$-deformed golden ratio has the smallest radius of convergence among all real numbers. The conjecture is proved for certain class of rational numbers and confirmed by a number of computer experiments. We also prove the explicit lower bounds for the radius of convergence for the $q$-deformed convergents of golden and silver ratios.
We explain the notion of "$q$-deformed real numbers" introduced in our previous work and overview their main properties. We will also introduce $q$-deformed Conway-Coxeter friezes.
We develop further the theory of $q$-deformations of real numbers introduced by Morier-Genoud and Ovsienko, and focus in particular on the class of real quadratic irrationals. Our key tool is a $q$-deformation of the modular group $PSL_q(2,\mathbb{Z})$. The action of the modular group by Möbius transformations commutes with the $q$-deformations. We prove that the traces of the elements of $PSL_q(2,\mathbb{Z})$ are palindromic polynomials with positive coefficients. These traces appear in the explicit expressions of the $q$-deformed quadratic irrationals.
We count the number of Coxeter's friezes over a finite field. Our method uses geometric realizations of the spaces of friezes in a certain completion of the classical moduli space $\mathcal{M}_{0,n}$ allowing repeated points in the configurations. Counting points in the completed moduli space over a finite field is related to the enumeration problem of counting partitions of cyclically ordered set of points into subsets containing no consecutive points. In Appendix we provide an elementary solution for this enumeration problem.
We introduce a notion of $q$-deformed rational numbers and $q$-deformed continued fractions. A $q$-deformed rational is encoded by a triangulation of a polygon and can be computed recursively. The recursive formula is analogous to the $q$-deformed Pascal identitiy for the Gaussian binomial coefficients, but the Pascal triangle is replaced by the Farey graph. The coefficients of the polynomials defining the $q$-rational count quiver subrepresentations of the maximal indecomposable representation of the graph dual to the triangulation. Several other properties, such as total positivity properties, $q$-deformation of the Farey graph, matrix presentations and $q$-continuants are given, as well as a relation to the Jones polynomial of rational knots.
We introduce a new class of friezes which is related to symplectic geometry. On the algebraic and combinatrics sides, this variant of friezes is related to the cluster algebras involving the Dynkin diagrams of type ${\rm C}_{2}$ and ${\rm A}_{m}$. On the geometric side, they are related to the moduli space of Lagrangian configurations of points in the 4-dimensional symplectic space introduced in [Conley C.H., Ovsienko V., Math. Ann. 375 (2019), 1105-1145]. Symplectic friezes share similar combinatorial properties to those of Coxeter friezes and SL-friezes.
We associate a formal power series with integer coefficients to a positive real number, we interpret this series as a "$q$-analogue of a real." The construction is based on the notion of $q$-deformed rational number introduced in arXiv:1812.00170. Extending the construction to negative real numbers, we obtain certain Laurent series.
We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections appear when extending these theorems for elements of the modular group $PSL(2,\mathbb{Z})$. These polygon dissections are interpreted as walks in the Farey tessellation. The combinatorial model of continued fractions can be further developed to obtain a canonical presentation of elements of $PSL(2,\mathbb{Z})$.
We consider tangent cones of Schubert varieties in the complete flag variety, and investigate the problem when the tangent cones of two different Schubert varieties coincide. We give a sufficient condition for such coincidence, and formulate a conjecture that provides a necessary condition. In particular, we show that all Schubert varieties corresponding to the Coxeter elements of the Weyl group have the same tangent cone. Our main tool is the notion of pillar entries in the rank matrix counting the dimensions of the intersections of a given flag with the standard one. This notion is a version of Fulton's essential set. We calculate the dimension of a Schubert variety in terms of the pillar entries of the rank matrix.
Frieze patterns of numbers, introduced in the early 70's by Coxeter, are currently attracting much interest due to connections with the recent theory of cluster algebras. The present paper aims to review the original work of Coxeter and the new developments around the notion of frieze, focusing on the representation theoretic, geometric and combinatorial approaches.
We introduce a supersymmetric analog of the classical Coxeter frieze patterns. Our approach is based on the relation with linear difference operators. We define supersymmetric analogs of linear difference operators called Hill's operators. The space of these "superfriezes" is an algebraic supervariety, which is isomorphic to the space of supersymmetric second order difference equations, called Hill's equations.
The notion of $SL_2$-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic $SL_2$-tilings that contain a rectangular domain of positive integers. Every such $SL_2$-tiling corresponds to a pair of frieze patterns and a unimodular $2\times2$-matrix with positive integer coefficients. We relate this notion to triangulated $n$-gons in the Farey graph.
We consider a family, $\mathcal{F}$, of subsets of an $n$-set such that the cardinality of the symmetric difference of any two elements $F,F'\in\mathcal{F}$ is not a multiple of 4. We prove that the maximal size of $\mathcal{F}$ is bounded by $2n$, unless $n\equiv{}3\mod4$ when it is bounded by $2n+2$. Our method uses cubic forms on $\mathbb{F}_2^n$ and the Hurwitz-Radon theory of square identities. We also apply this theory to obtain some information about boolean cubic forms and so-called additive quadruples.