12 research outputs found
Shifted Quasi-Symmetric Functions and the Hopf algebra of peak functions
In his work on P-partitions, Stembridge defined the algebra of peak functions
Pi, which is both a subalgebra and a retraction of the algebra of
quasi-symmetric functions. We show that Pi is closed under coproduct, and
therefore a Hopf algebra, and describe the kernel of the retraction. Billey and
Haiman, in their work on Schubert polynomials, also defined a new class of
quasi-symmetric functions --- shifted quasi-symmetric functions --- and we show
that Pi is strictly contained in the linear span Xi of shifted quasi-symmetric
functions. We show that Xi is a coalgebra, and compute the rank of the n-th
graded component.Comment: 9 pages, 4 eps figures, uses epsf.sty. to be presented at FPSAC99 in
Barcelona by second autho
Non-commutative Pieri operators on posets
We consider graded representations of the algebra NC of noncommutative
symmetric functions on the Z-linear span of a graded poset P. The matrix
coefficients of such a representation give a Hopf morphism from a Hopf algebra
HP generated by the intervals of P to the Hopf algebra of quasi-symmetric
functions. This provides a unified construction of quasi-symmetric generating
functions from different branches of algebraic combinatorics, and this
construction is useful for transferring techniques and ideas between these
branches. In particular we show that the (Hopf) algebra of Billera and Liu
related to Eulerian posets is dual to the peak (Hopf) algebra of Stembridge
related to enriched P-partitions, and connect this to the combinatorics of the
Schubert calculus for isotropic flag manifolds.Comment: LaTeX 2e, 22 pages Minor corrections, updated references. Complete
and final version, to appear in issue of J. Combin. Th. Ser. A dedicated to
G.-C. Rot
Combinatorial Hopf algebras and generalized Dehn-Sommerville relations
A combinatorial Hopf algebra is a graded connected Hopf algebra over a field
equipped with a character (multiplicative linear functional) . We show that the terminal object in the category of combinatorial Hopf
algebras is the algebra 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 possesses two
canonical Hopf subalgebras on which the character 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 , the generalized Dehn-Sommerville relations are the
Bayer-Billera relations and the odd subalgebra is the peak Hopf algebra of
Stembridge. We prove that 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.Comment: 34 page