| 1. |
- Ardila, Federico, et al.
(författare)
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The equivariant Ehrhart theory of the permutahedron
- 2020
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Ingår i: Proceedings of the American Mathematical Society. - : American Mathematical Society (AMS). - 0002-9939 .- 1088-6826. ; 148:12, s. 5091-5107
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Tidskriftsartikel (refereegranskat)abstract
- Equivariant Ehrhart theory enumerates the lattice points in a polytope with respect to a group action. Answering a question of Stapledon, we describe the equivariant Ehrhart theory of the permutahedron, and we prove his Effectiveness Conjecture in this special case.
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| 2. |
- Elia, Sophia, et al.
(författare)
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Techniques in Equivariant Ehrhart Theory
- 2023
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Ingår i: Annals of Combinatorics. - : Springer Nature. - 0218-0006 .- 0219-3094.
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Tidskriftsartikel (refereegranskat)abstract
- Equivariant Ehrhart theory generalizes the study of lattice point enumeration to also account for the symmetries of a polytope under a linear group action. We present a catalogue of techniques with applications in this field, including zonotopal decompositions, symmetric triangulations, combinatorial interpretation of the h∗ -polynomial, and certificates for the (non)existence of invariant nondegenerate hypersurfaces. We apply these methods to several families of examples including hypersimplices, orbit polytopes, and graphic zonotopes, expanding the library of polytopes for which their equivariant Ehrhart theory is known.
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| 3. |
- Eur, Christopher, et al.
(författare)
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The universal valuation of Coxeter matroids
- 2021
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Ingår i: Bulletin of the London Mathematical Society. - : Wiley. - 0024-6093 .- 1469-2120. ; 53:3, s. 798-819
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Tidskriftsartikel (refereegranskat)abstract
- Coxeter matroids generalize matroids just as flag varieties of Lie groups generalize Grassmannians. Valuations of Coxeter matroids are functions that behave well with respect to subdivisions of a Coxeter matroid into smaller ones. We compute the universal valuative invariant of Coxeter matroids. A key ingredient is the family of Coxeter Schubert matroids, which correspond to the Bruhat cells of flag varieties. In the process, we compute the universal valuation of generalized Coxeter permutohedra, a larger family of polyhedra that model Coxeter analogues of combinatorial objects such as matroids, clusters, and posets.
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| 4. |
- Supina, Mariel
(författare)
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The Hopf monoid of orbit polytopes
- 2020
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Ingår i: Journal of Combinatorics. - : International Press of Boston. - 2156-3527 .- 2150-959X. ; 11:4, s. 575-601
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Tidskriftsartikel (refereegranskat)abstract
- Many families of combinatorial objects have a Hopf monoid structure. Aguiar and Ardila introduced the Hopf monoid of generalized permutahedra and showed that it contains various other notable combinatorial families as Hopf submonoids, including graphs, posets, and matroids. We introduce the Hopf monoid of orbit polytopes, which is generated by the generalized permutahedra that are invariant under the action of the symmetric group. We show that modulo normal equivalence, these polytopes are in bijection with integer compositions. We interpret the Hopf structure through this lens, and we show that applying the first Fock functor to this Hopf monoid gives a Hopf algebra of compositions. We describe the character group of the Hopf monoid of orbit polytopes in terms of noncommutative symmetric functions, and we give a combinatorial interpretation of the basic character and its polynomial invariant.
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