Symplectic Hecke eigenbases from Ehrhart polynomials
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866916845141884928 |
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| author | Alfes, Claudia Maglione, Joshua Voll, Christopher |
| author_facet | Alfes, Claudia Maglione, Joshua Voll, Christopher |
| contents | For $n\in\mathbb{N}$ and $\ell\in\{0,1,\dots,n\}$, we consider the function extracting the $\ell$th coefficient of the Ehrhart polynomials of lattice polytopes in $\mathbb{R}^n$. These functions form a basis of the space of unimodular invariant valuations. We show that, in even dimensions, these functions are in fact simultaneous symplectic Hecke eigenfunctions. We leverage this and apply the theory of spherical functions and their associated zeta functions to prove analytic, asymptotic, and combinatorial results about the arithmetic functions averaging $\ell$th Ehrhart coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_11728 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Symplectic Hecke eigenbases from Ehrhart polynomials Alfes, Claudia Maglione, Joshua Voll, Christopher Combinatorics Group Theory Number Theory 20C08, 52B20, 43A90, 05A15, 11M41 For $n\in\mathbb{N}$ and $\ell\in\{0,1,\dots,n\}$, we consider the function extracting the $\ell$th coefficient of the Ehrhart polynomials of lattice polytopes in $\mathbb{R}^n$. These functions form a basis of the space of unimodular invariant valuations. We show that, in even dimensions, these functions are in fact simultaneous symplectic Hecke eigenfunctions. We leverage this and apply the theory of spherical functions and their associated zeta functions to prove analytic, asymptotic, and combinatorial results about the arithmetic functions averaging $\ell$th Ehrhart coefficients. |
| title | Symplectic Hecke eigenbases from Ehrhart polynomials |
| topic | Combinatorics Group Theory Number Theory 20C08, 52B20, 43A90, 05A15, 11M41 |
| url | https://arxiv.org/abs/2507.11728 |