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Register a new userThe Hopf monoid of hypergraphs and its sub-monoids: basic invariant and reciprocity theorem
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In arXiv:1709.07504 Ardila and Aguiar give a Hopf monoid structure on hypergraphs as well as a general construction of polynomial invariants on Hopf monoids. Using these results, we define in this paper a new polynomial invariant on hypergraphs. We give a combinatorial interpretation of this invariant on negative integers which leads to a reciprocity theorem on hypergraphs. Finally, we use this invariant to recover well-known invariants on other combinatorial objects (graphs, simplicial complexes, building sets etc) as well as the associated reciprocity theorems.
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In arXiv:1709.07504 Aguiar and Ardila give a Hopf monoid structure on hypergraphs as well as a general construction of polynomial invariants on Hopf monoids. Using these results, we define in this paper a new polynomial invariant on hypergraphs. We give a combinatorial interpretation of this invariant on negative integers which leads to a reciprocity theorem on hypergraphs. Finally, we use this invariant to recover well-known invariants on other combinatorial objects (graphs, simplicial complexes, building sets etc) as well as the associated reciprocity theorems.
From a recent paper, we recall the Hopf monoid structure on the supercharacters of the unipotent uppertriangular groups over a finite field. We give cancelation free formula for the antipode applied to the bases of class functions and power sum functions, giving new cancelation free formulae for the standard Hopf algebra of supercharacters and symmetric functions in noncommuting variables. We also give partial results for the antipode on the supercharacter basis, and explicitly describe the primitives of this Hopf monoid.
Many combinatorial Hopf algebras $H$ in the literature are the functorial image of a linearized Hopf monoid $bf H$. That is, $H={mathcal K} ({bf H})$ or $H=overline{mathcal K} ({bf H})$. Unlike the functor $overline{mathcal K}$, the functor ${mathcal K}$ applied to ${bf H}$ may not preserve the antipode of ${bf H}$. In this case, one needs to consider the larger Hopf monoid ${bf L}times{bf H}$ to get $H={mathcal K} ({bf H})=overline{mathcal K}({bf L}times{bf H})$ and study the antipode in ${bf L}times{bf H}$. One of the main results in this paper provides a cancelation free and multiplicity free formula for the antipode of ${bf L}times{bf H}$. From this formula we obtain a new antipode formula for $H$. We also explore the case when ${bf H}$ is commutative and cocommutative. In this situation we get new antipode formulas that despite of not being cancelation free, can be used to obtain one for $overline{mathcal K}({bf H})$ in some cases. We recover as well many of the well-known cancelation free formulas in the literature. One of our formulas for computing the antipode in ${bf H}$ involves acyclic orientations of hypergraphs as the central tool. In this vein, we obtain polynomials analogous to the chromatic polynomial of a graph, and also identities parallel to Stanleys (-1)-color theorem. One of our examples introduces a {it chromatic} polynomial for permutations which counts increasing sequences of the permutation satisfying a pattern. We also study the statistic obtained after evaluating such polynomial at $-1$. Finally, we sketch $q$ deformations and geometric interpretations of our results. This last part will appear in a sequel paper in joint work with J. Machacek.
We study Cohen-Macaulay Hopf monoids in the category of species. The goal is to apply techniques from topological combinatorics to the study of polynomial invariants arising from combinatorial Hopf algebras. Given a polynomial invariant arising from a linearized Hopf monoid, we show that under certain conditions it is the Hilbert polynomial of a relative simplicial complex. If the Hopf monoid is Cohen-Macaulay, we give necessary and sufficient conditions for the corresponding relative simplicial complex to be relatively Cohen-Macaulay, which implies that the polynomial has a nonnegative $h$-vector. We apply our results to the weak and strong chromatic polynomials of acyclic mixed graphs, and the order polynomial of a double poset.
We build, from the collection of all groups of unitriangular matrices, Hopf monoids in Joyals category of species. Such structure is carried by the collection of class function spaces on those groups, and also by the collection of superclass function spaces, in the sense of Diaconis and Isaacs. Superclasses of unitriangular matrices admit a simple description from which we deduce a combinatorial model for the Hopf monoid of superclass functions, in terms of the Hadamard product of the Hopf monoids of linear orders and of set partitions. This implies a recent result relating the Hopf algebra of superclass functions on unitriangular matrices to symmetric functions in noncommuting variables. We determine the algebraic structure of the Hopf monoid: it is a free monoid in species, with the canonical Hopf structure. As an application, we derive certain estimates on the number of conjugacy classes of unitriangular matrices.
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