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The equivariant Ehrhart theory of the permutahedron. (English) Zbl 1472.52016

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.

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.)

Citations:

Zbl 1218.52014
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References:

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