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Register a new userOn Cohen-Macaulay Hopf monoids in species
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Authors
Jacob White
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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.
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Let $G=(V,E)$ be a graph. If $G$ is a Konig graph or $G$ is a graph without 3-cycles and 5-cycle, we prove that the following conditions are equivalent: $Delta_{G}$ is pure shellable, $R/I_{Delta}$ is Cohen-Macaulay, $G$ is unmixed vertex decomposable graph and $G$ is well-covered with a perfect matching of Konig type $e_{1},...,e_{g}$ without square with two $e_i$s. We characterize well-covered graphs without 3-cycles, 5-cycles and 7-cycles. Also, we study when graphs without 3-cycles and 5-cycles are vertex decomposable or shellable. Furthermore, we give some properties and relations between critical, extendables and shedding vertices. Finally, we characterize unicyclic graphs with each one of the following properties: unmixed, vertex decomposable, shellable and Cohen-Macaulay.
Let $H$ be a semisimple Hopf algebra, and let $R$ be a noetherian left $H$-module algebra. If $R/R^H$ is a right $H^*$-dense Galois extension, then the invariant subalgebra $R^H$ will inherit the AS-Cohen-Macaulay property from $R$ under some mild conditions, and $R$, when viewed as a right $R^H$-module, is a Cohen-Macaulay module. In particular, we show that if $R$ is a noetherian complete semilocal algebra which is AS-regular of global dimension 2 and $H=operatorname{bf k} G$ for some finite subgroup $Gsubseteq Aut(R)$, then all the indecomposable Cohen-Macaulay module of $R^H$ is a direct summand of $R_{R^H}$, and hence $R^H$ is Cohen-Macaulay-finite, which generalizes a classical result for commutative rings. The main tool used in the paper is the extension groups of objects in the corresponding quotient categories.
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.
We study invariants and quotient categories of fixed subrings of Artin-Schelter regular algebras under Hopf algebra actions.
Given a projective algebraic set X, its dual graph G(X) is the graph whose vertices are the irreducible components of X and whose edges connect components that intersect in codimension one. Hartshornes connectedness theorem says that if (the coordinate ring of) X is Cohen-Macaulay, then G(X) is connected. We present two quantitative variants of Hartshornes result: 1) If X is a Gorenstein subspace arrangement, then G(X) is r-connected, where r is the Castelnuovo-Mumford regularity of X. (The bound is best possible; for coordinate arrangements, it yields an algebraic extension of Balinskis theorem for simplicial polytopes.) 2) If X is a canonically embedded arrangement of lines no three of which meet in the same point, then the diameter of the graph G(X) is not larger than the codimension of X. (The bound is sharp; for coordinate arrangements, it yields an algebraic expansion on the recent combinatorial result that the Hirsch conjecture holds for flag normal simplicial complexes.)
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