Ardila, Federico; Supina, Mariel; Vindas-Meléndez, Andrés R. The equivariant Ehrhart theory of the permutahedron. (English) Zbl 1472.52016 Sémin. Lothar. Comb. 84B, Article 83, 12 p. (2020). In [A. Stapledon, Adv. Math. 226, No. 4, 3622–3654 (2011; zbl 1218.52014)] Stapledon introduced equivariant Ehrhart theory, a variant of Ehrhart theory that takes group actions into account. For a lattice polytope \(P\) whose vertices lie in the lattice \(M\) and a group \(G\) acting on \(M\), one can define the equivariant \(H^*\)-series \(H^*[z]\) which can be written as \(\sum_{i\geq 0} H_i^*z^i\) for appropriate virtual characters \(H_i^*\). Stapledon asks whether or not this series is effective, i.e, whether all the \(H_i^*\) are characters of representations of \(G\), and proposes the effectiveness conjecture which states that the effectiveness of the equivariant \(H^*\)-series is equivalent to two other properties, namely (i) the toric variety of \(P\) admits a \(G\)-invariant non-degenerate hypersurface,(ii) the equivariant \(H^*\)-series is a polynomial. It is already known that (i) is a sufficient and (ii) is a necessary condition.The present paper proves the effectiveness conjecture and three minor conjectures in the case of permutahedra under the action of the symmetric group. Reviewer: Max Kölbl (Leipzig) MSC: 52B20 Lattice polytopes in convex geometry (including relations with commutative algebra and algebraic geometry) 05E10 Combinatorial aspects of representation theory 05E14 Combinatorial aspects of algebraic geometry 20C30 Representations of finite symmetric groups 51M20 Polyhedra and polytopes; regular figures, division of spaces 52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.) Keywords:Ehrhart theory; permutahedron; quasipolynomial; symmetric group; representation theory; zonotope Citations:Zbl 1218.52014 PDFBibTeX XMLCite \textit{F. Ardila} et al., Sémin. Lothar. Comb. 84B, Article 83, 12 p. (2020; Zbl 1472.52016) Full Text: arXiv Link References: [1] F. Ardila, M. Beck, and J. McWhirter. “Ehrhart quasipolynomials of Coxeter permutahedra”. In preparation. 2019. [2] F. Ardila, A. Schindler, and A. R. Vindas-Meléndez. “The equivariant volumes of the permutahedron”.Discrete and Computational Geometry(2019), pp. 1-18.Link. · Zbl 1436.51019 [3] M. Beck and S. Robins.Computing the Continuous Discretely. Undergraduate Texts in Mathematics. Integer-point enumeration in polyhedra. Springer, New York, 2007, pp. xviii+226. · Zbl 1114.52013 [4] E. Ehrhart. “Sur les polyèdres rationnels homothétiques àndimensions”.C. R. Acad. Sci. Paris254(1962), pp. 616-618. · Zbl 0100.27601 [5] A. G. Khovanskii. “Newton polyhedra and toroidal varieties”.Functional analysis and its applications11.4 (1977), pp. 289-296.Link. · Zbl 0445.14019 [6] J. McWhirter. “Ehrhart quasipolynomials of Coxeter permutahedra”. Masters thesis. 2019. [7] N. Perminov and S. Shakirov. “Discriminants of symmetric polynomials”. Preprint. 2009. arXiv:0910.5757. [8] G. C. Shephard. “Combinatorial properties of associated zonotopes”.Canad. J. Math.26 (1974), pp. 302-321.Link. · Zbl 0287.52005 [9] R. P. Stanley.A zonotope associated with graphical degree sequences. Vol. 4. DIMACS Ser. Discrete Math. Theoret. Comput. Sci. Amer. Math. Soc., Providence, RI, 1991, pp. 555-570. Link. · Zbl 0737.05057 [10] A. This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.