[Paper Review] Introducing supersymmetric frieze patterns and linear difference operators

Sophie Morier-Genoud, Valentin Ovsienko+1 more|arXiv (Cornell University)|Jan 29, 2015

Algebraic structures and combinatorial models18 references6 citations

TL;DR

This paper introduces supersymmetric frieze patterns as a super-analog of classical Coxeter friezes, using supersymmetric linear difference operators and the supergroup OSp(1|2). It establishes a direct isomorphism between the space of superfriezes and the supervariety of supersymmetric Hill’s equations, proving a Laurent phenomenon and expressing supercontinuants via Berezinians.

ABSTRACT

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.

Motivation & Objective

• To develop a supersymmetric analog of classical Coxeter frieze patterns using superalgebraic structures.
• To define supersymmetric linear difference operators, particularly supersymmetric Hill’s operators with (anti)periodic solutions.
• To establish a geometric and algebraic correspondence between superfriezes and supersymmetric second-order difference equations.
• To prove the Laurent phenomenon in superfriezes and express supercontinuants via Berezinians.
• To lay the foundation for cluster superalgebras by introducing a new class of algebraic supervarieties.

Proposed method

• Introduces a supersymmetric shift operator 𝔗 satisfying 𝔗² = −T, discretizing the odd supersymmetric vector field D = ∂ξ − ξ∂x.
• Defines supersymmetric discrete Sturm-Liouville operators and their associated Hill’s equations with periodic coefficients.
• Constructs superfriezes using a modified Coxeter frieze rule, replacing SL₂ with the supergroup OSp(1|2).
• Establishes a bijection between generic superfriezes and solutions of supersymmetric Hill’s equations via recurrence relations.
• Expresses supercontinuants using determinants and Berezinians, with explicit matrix formulas involving even and odd entries.
• Proves the Laurent phenomenon via induction on recurrence relations and determinant identities.

Experimental results

Research questions

• RQ1 How can classical Coxeter frieze patterns be generalized to a supersymmetric setting using superalgebraic structures?
• RQ2 What is the role of the supergroup OSp(1|2) in defining the structure of superfriezes?
• RQ3 How do supersymmetric Hill’s equations relate to the geometry of the supervariety 𝒪Eₙ?
• RQ4 Can the Laurent phenomenon, known in classical friezes, be extended to the supercase?
• RQ5 What is the algebraic structure of supercontinuants in terms of Berezinians and matrix determinants?

Key findings

• The space of superfriezes is isomorphic to the supervariety 𝒪Eₙ of supersymmetric Hill’s equations, establishing a direct algebraic correspondence.
• The Laurent phenomenon holds for superfriezes, with all entries expressed as Laurent polynomials in the initial data.
• Supercontinuants K(a₁|β₁,β₁|…|aₙ|βₙ,βₙ) are expressed as Berezinians of a 2×2 block matrix with specific even and odd components.
• The determinant formula for supercontinuants is derived via recurrence and verified by induction, matching known integer sequences A077998, A006054, and A052534.
• The supersymmetric shift operator 𝔗 satisfies 𝔗² = −T, providing a discrete analog of the odd supersymmetric vector field D.
• The glide symmetry and periodicity of generic superfriezes are rigorously established, generalizing classical frieze properties to the supercase.