Stephanie van Willigenburg

Peak quasisymmetric functions and Eulerian enumeration

Abstract Via duality of Hopf algebras, there is a direct association between peak quasisymmetric functions and enumeration of chains in Eulerian posets. We study this association explicitly, showing that the notion of cd -index, long studied in the context of convex polytopes and Eulerian posets, arises as the dual basis to a natural basis of peak quasisymmetric functions introduced by Stembridge. Thus Eulerian posets having a nonnegative cd -index (for example, face lattices of convex polytopes) correspond to peak quasisymmetric functions having a nonnegative representation in terms of this basis. We diagonalize the operator that associates the basis of descent sets for all quasisymmetric functions to that of peak sets for the algebra of peak functions, and study the g -polynomial for Eulerian posets as an algebra homomorphism.

Noncommutative 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.

Shifted quasi-symmetric functions and the Hopf algebra of peak functions

In his work on P-partitions, Stembridge defined the algebra of peak functions Π, which is both a subalgebra and a retraction of the algebra of quasi-symmetric functions. We show that Π 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 Π is strictly contained in the linear span Ξ of shifted quasi-symmetric functions. We show that Ξ is a coalgebra, and compute the rank of the nth graded component.

Enumerative Properties of Ferrers Graphs

Abstract We define a class of bipartite graphs that correspond naturally with Ferrers diagrams. We give expressions for the number of spanning trees, the number of Hamiltonian paths when applicable, the chromatic polynomial and the chromatic symmetric function. We show that the linear coefficient of the chromatic polynomial is given by the excedance set statistic.

Towards a combinatorial classification of skew Schur functions

We present a single operation for constructing skew diagrams whose corresponding skew Schur functions are equal. This combinatorial operation naturally generalises and unifies all results of this type to date. Moreover, our operation suggests a closely related condition that we conjecture is necessary and sufficient for skew diagrams to yield equal skew Schur functions.

Positivity results on ribbon Schur function differences

There is considerable current interest in determining when the difference of two skew Schur functions is Schur positive. We consider the posets that result from ordering skew diagrams according to Schur positivity, before focussing on the convex subposets corresponding to ribbons. While the general solution for ribbon Schur functions seems out of reach at present, we determine necessary and sufficient conditions for multiplicity-free ribbons, i.e. those whose expansion as a linear combination of Schur functions has all coefficients either zero or one. In particular, we show that the poset that results from ordering such ribbons according to Schur positivity is essentially a product of two chains.

On tensor products of polynomial representations

We determine the necessary and sufficient combinatorial conditions for which the tensor product of two irreducible polynomial representations of GL(n, C) is isomorphic to another. As a con- sequencewediscoverfamiliesofLittlewood-Richardson coefficientsthatarenon-zero, and acondition on Schur non-negativity.

Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions

The classical Littlewood-Richardson rule is a rule for computing coefficients in many areas, and comes in many guises. In this paper we prove two Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions that are analogous to the famed version of the classical Littlewood-Richardson rule involving Yamanouchi words. Furthermore, both our rules contain this classical Littlewood-Richardson rule as a special case. We then apply our rules to combinatorially classify symmetric skew quasisymmetric Schur functions. This answers affirmatively a conjecture of Bessenrodt, Luoto and van Willigenburg.

A Multiplication Rule for the Descent Algebra of TypeD

Here we give a combinatorial interpretation of Solomons rule for multiplication in the descent algebra of Weyl groups of type

ON (ALMOST) EXTREME COMPONENTS IN KRONECKER PRODUCTS OF CHARACTERS OF THE SYMMETRIC GROUPS

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