Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials

Loïc Foissy (1)

1 LMPA - Laboratoire de Mathématiques Pures et Appliquées Joseph Liouville

To any poset or quasi-poset is attached a lattice polytope, whose Ehrhart polynomial we study from a Hopf-algebraic point of view. We use for this two interacting bialgebras on quasi-posets. The Ehrhart polynomial defines a Hopf algebra morphism taking its values in Q[X]; we deduce from the interacting bialgebras an algebraic proof of the duality principle, a generalization and a new proof of a result on B-series due to Whright and Zhao, using a monoid of characters on quasi-posets, and a generalization of Faulhaber's formula. We also give non-commutative versions of these results: polynomials are replaced by packed words. We obtain in particular a non-commutative duality principle.

Mots clés: Ehrhart polynomials; Quasi-posets; Characters monoids; Interacting bialgebras

Domaines: Anneaux et algèbres [math.RA]

Loïc Foissy. Commutative and non-commutative bialgebras of quasi-posets and applications to Ehrhart polynomials. Advances in Pure and Applied Mathematics, 2019, 10 (1), pp.27-63. ⟨10.1515/apam-2016-0051⟩. ⟨hal-01321986v2⟩