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      A matroid-friendly basis for the quasisymmetric functions

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          Abstract

          A new Z-basis for the space of quasisymmetric functions (QSym, for short) is presented. It is shown to have nonnegative structure constants, and several interesting properties relative to the space of quasisymmetric functions associated to matroids by the Hopf algebra morphism (F) of Billera, Jia, and Reiner. In particular, for loopless matroids, this basis reflects the grading by matroid rank, as well as by the size of the ground set. It is shown that the morphism F is injective on the set of rank two matroids, and that decomposability of the quasisymmetric function of a rank two matroid mirrors the decomposability of its base polytope. An affirmative answer is given to the Hilbert basis question raised by Billera, Jia, and Reiner.

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          A Course in Convexity

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            Combinatorial Hopf algebras and generalized Dehn-Sommerville relations

            A combinatorial Hopf algebra is a graded connected Hopf algebra over a field \(F\) equipped with a character (multiplicative linear functional) \(\zeta:H\to F\). We show that the terminal object in the category of combinatorial Hopf algebras is the algebra \(QSym\) of quasi-symmetric functions; this explains the ubiquity of quasi-symmetric functions as generating functions in combinatorics. We illustrate this with several examples. We prove that every character decomposes uniquely as a product of an even character and an odd character. Correspondingly, every combinatorial Hopf algebra \((H,\zeta)\) possesses two canonical Hopf subalgebras on which the character \(\zeta\) is even (respectively, odd). The odd subalgebra is defined by certain canonical relations which we call the generalized Dehn-Sommerville relations. We show that, for \(H=QSym\), the generalized Dehn-Sommerville relations are the Bayer-Billera relations and the odd subalgebra is the peak Hopf algebra of Stembridge. We prove that \(QSym\) is the product (in the categorical sense) of its even and odd Hopf subalgebras. We also calculate the odd subalgebras of various related combinatorial Hopf algebras: the Malvenuto-Reutenauer Hopf algebra of permutations, the Loday-Ronco Hopf algebra of planar binary trees, the Hopf algebras of symmetric functions and of non-commutative symmetric functions.
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              Chirurgie des grassmanniennes

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                Author and article information

                Journal
                05 April 2007
                2007-11-16
                Article
                10.1016/j.jcta.2007.10.003
                0704.0836
                83835ad9-ac25-4014-8071-d929c956725f
                History
                Custom metadata
                05B35, 52B40
                Journal of Combinatorial Theory, Series A 115 (2008) 777-798
                25 pages; exposition tightened, typos corrected; to appear in the Journal of Combinatorial Theory, Series A
                math.CO

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