[Paper Review] On a Relation between Spectral Theory of Lens Spaces and Ehrhart Theory

Hossein Mohades, B. Honari|arXiv (Cornell University)|Jan 17, 2016

Advanced Combinatorial Mathematics18 references3 citations

TL;DR

This paper establishes a connection between the spectral theory of lens spaces and Ehrhart theory by associating a rational simplex to each lens space and showing that isospectrality implies equality in the number of lattice points in dilated polytopes and in global sections of line bundles on associated toric varieties. The key contribution is a direct proof of Lauret, Miatello, and Rossetti's theorem on isospectral lens spaces using harmonic polynomial representations of SO(n).

ABSTRACT

In this article Ehrhart quasi-polynomials of simplices are employed to determine isospectral lens spaces in terms of a finite set of numbers. Using the natural lattice associated with a lens space the associated toric variety of a lens space is introduced. It is proved that if two lens spaces are isospectral then the dimension of global sections of powers of a natural line bundle on these two toric varieties are equal and they have the same general intersection number. Also, harmonic polynomial representation of the group SO(n) is used to provide a direct proof for a theorem of Lauret, Miatello and Rossetti on isospectrality of lens spaces.

Motivation & Objective

• To establish a geometric and algebraic bridge between isospectrality in lens spaces and Ehrhart theory of rational polytopes.
• To define a natural toric variety associated with each lens space using its underlying lattice and rational simplex.
• To prove that isospectral lens spaces have equal dimensions of global sections of powers of a natural line bundle on their associated toric varieties.
• To provide a direct proof of the isospectrality criterion for lens spaces using harmonic polynomial representations of SO(n).
• To generalize the understanding of isospectrality beyond representation-theoretic methods by linking it to combinatorial geometry and algebraic geometry.

Proposed method

• Associate a rational simplex to each lens space using its natural lattice structure.
• Define the Ehrhart quasi-polynomial of this simplex to count lattice points in dilated copies.
• Construct the toric variety associated with the simplex via the associated fan and line bundle.
• Use harmonic homogeneous polynomials on R^n to realize irreducible representations of SO(n) and compute spectral multiplicities.
• Apply the Bernstein-Kouchnirenko theorem to relate the degree of the toric variety to the volume of the polytope.
• Leverage the fact that isospectrality implies equality in the number of lattice points in dilated polytopes, hence equality in global sections of line bundles.

Experimental results

Research questions

• RQ1 Can isospectrality of lens spaces be characterized via the Ehrhart theory of an associated rational simplex?
• RQ2 Is there a natural toric variety associated with a lens space such that isospectrality is equivalent to equality in the dimensions of global sections of powers of a natural line bundle?
• RQ3 Can the isospectrality criterion for lens spaces be proven directly using harmonic polynomial representations of SO(n)?
• RQ4 How do the global sections of line bundles on the associated toric varieties reflect spectral invariants of lens spaces?
• RQ5 Can the intersection numbers and degrees of the associated toric varieties be used to characterize isospectral lens spaces?

Key findings

• If two lens spaces are isospectral, then the dimensions of the spaces of global sections of the k-th power of the natural line bundle on their associated toric varieties are equal for all k ∈ ℕ.
• The degrees of the associated toric varieties are equal, as they are determined by the leading coefficient of the Hilbert polynomial, which corresponds to the volume of the rational simplex.
• The Ehrhart quasi-polynomial of the simplex associated with a lens space counts the number of lattice points in dilated copies, which directly corresponds to the multiplicity of eigenvalues in the spectrum.
• The proof of the isospectrality criterion by Lauret, Miatello, and Rossetti is re-derived using harmonic polynomial representations of SO(n), providing a direct algebraic-geometric argument.
• The natural line bundle on the toric variety associated with a lens space has global sections whose dimensions are given by the Ehrhart quasi-polynomial evaluated at k.
• The general intersection number of the toric variety is preserved under isospectrality, reflecting the invariance of the top-dimensional intersection number under spectral equivalence.