On a relation between spectral theory of lens spaces and Ehrhart theory |
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Institution: | Department of Mathematics and Computer Science, Amirkabir University of Technology, Tehran, Iran |
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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() is used to provide a more elementary proof for a theorem of Lauret, Miatello and Rossetti on isospectrality of lens spaces. |
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Keywords: | Lens space Laplace–Beltrami operator Isospectrality Harmonic polynomial Ehrhart quasi-polynomial Rational convex polytope Toric variety Divisor Line bundle |
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