Explicit polynomial generators for the ring of quasi-symmetric functions over the integers

We describe generating functions for several important families of classical symmetric functions and shifted Schur functions. The approach is originated from vertex operator realization of symmetric functions and offers a unified method to treat various families of symmetric functions and their shifted analogues.

We extend past results on a family of formal power series $K_{n, \Lambda}$, parameterized by $n$ and $\Lambda \subseteq [n]$, that largely resemble quasisymmetric functions. This family of functions was conjectured to have the property that the product $K_{n, \Lambda}K_{m, \Omega}$ of any two functions $K_{n, \Lambda}$ and $K_{m, \Omega}$ from the family can be expressed as a linear combination of other functions from the family. In this paper, we show that this is indeed the...

We create several families of bases for the symmetric polynomials. From these bases we prove that certain Schur symmetric polynomials form a basis for quotients of symmetric polynomials that generalize the cohomology and the quantum cohomology of the Grassmannian. Our work also provides an alternative proof of a result due to Grinberg.

Adem and Reichstein introduced the ideal of truncated symmetric polynomials to present the permutation invariant subring in the cohomology of a finite product of projective spaces. Building upon their work, I describe a generating set of the ideal of truncated symmetric polynomials in arbitrary positive characteristic, and offer a conjecture for minimal generators.

In this note we obtain numerous new bases for the algebra of symmetric functions whose generators are chromatic symmetric functions. More precisely, if $\{ G_ k \}_{k\geq 1}$ is a set of connected graphs such that $G_k$ has $k$ vertices for each $k$, then the set of all chromatic symmetric functions $\{ X_{G_ k} \}_{k\geq 1}$ generates the algebra of symmetric functions. We also obtain explicit expressions for the generators arising from complete graphs, star graphs, path gra...

A tropical polynomial in nr variables divided into blocks of r variables each, is r-symmetric, if it is invariant under the action of Sn that permutes the blocks. For r=1 we call these tropical polynomials symmetric. We can define r-symmetric and symmetric rational functions in a similar manner. In this paper we identify generators for the sets of symmetric tropical polynomials and rational functions. While r-symmetric tropical polynomials are not finitely generated, we show ...

A systematic procedure for generating certain identities involving elementary symmetric functions is proposed. These identities, as particular cases, lead to new identities for binomial and q-binomial coefficients.

We uncover the structure of the space of symmetric functions in non-commutative variables by showing that the underlined Hopf algebra is both free and co-free. We also introduce the Hopf algebra of quasi-symmetric functions in non-commutative variables and define the product and coproduct on the monomial basis of this space and show that this Hopf algebra is free and cofree. In the process of looking for bases which generate the space we define orders on the set partitions an...

This article is devoted to the study of several algebras which are related to symmetric functions, and which admit linear bases labelled by various combinatorial objects: permutations (free quasi-symmetric functions), standard Young tableaux (free symmetric functions) and packed integer matrices (matrix quasi-symmetric functions). Free quasi-symmetric functions provide a kind of noncommutative Frobenius characteristic for a certain category of modules over the 0-Hecke algebra...

We introduce a new basis of the non-commutative symmetric functions whose commutative images are Schur functions. Dually, we build a basis of the quasi-symmetric functions which expand positively in the fundamental quasi-symmetric functions and decompose Schur functions. We then use the basis to construct a non-commutative lift of the Hall-Littlewood symmetric functions with similar properties to their commutative counterparts.