Related papers: Schur-positivity in a Square

The classical algebra $\Lambda$ of symmetric functions has a remarkable deformation $\Lambda^*$, which we call the algebra of shifted symmetric functions. In the latter algebra, there is a distinguished basis formed by shifted Schur…

Cylindric skew Schur functions, a generalization of skew Schur functions, are closely related to the famous problem finding a combinatorial formula for the 3-point Gromov-Witten invariants of Grassmannian. In this paper, we prove cylindric…

It is a classical fundamental result that Schur-positive specializations of the ring of symmetric functions are characterized via totally positive functions whose parametrization describes the Edrei-Thoma theorem. In this paper we study…

We consider a type of divided symmetrization $\overrightarrow{D}_{\lambda,G}$ where $\lambda$ is a nonincreasing partition on $n$ and where $G$ is a graph. We discover that in the case where $\lambda$ is a hook shape partition with first…

A combinatorial expression for the coefficient of the Schur function $s_{\lambda}$ in the expansion of the plethysm $p_{n/d}^d \circ s_{\mu}$ is given for all $d$ dividing $n$ for the cases in which $n=2$ or $\lambda$ is rectangular. In…

We introduce new families of cylindric symmetric functions as subcoalgebras in the ring of symmetric functions $\Lambda$ (viewed as a Hopf algebra) which have non-negative structure constants. Combinatorially these cylindric symmetric…

The machinery of noncommutative Schur functions provides a general tool for obtaining Schur expansions for combinatorially defined symmetric functions. We extend this approach to a wider class of symmetric functions, explore its strengths…

When $\lambda$ is a partition, the specialized non-symmetric Macdonald polynomial $E_{\lambda}(x;q;0)$ is symmetric and related to a modified Hall--Littlewood polynomial. We show that whenever all parts of the integer partition $\lambda$ is…

Some new relations on skew Schur function differences are established both combinatorially using Sch\"utzenberger's jeu de taquin, and algebraically using Jacobi-Trudi determinants. These relations lead to the conclusion that certain…

We introduce partially defined Schur multipliers and obtain necessary and sufficient conditions for the existence of extensions to fully defined positive Schur multipliers, in terms of operator systems canonically associated with their…

FPSAC 2013 Extended Abstract. We introduce a new basis of the non-commutative symmetric functions whose elements have Schur functions as their commutative images. Dually, we build a basis of the quasi-symmetric functions which expand…

We give a new interpretation of the shifted Littlewood-Richardson coefficients $f_{\lambda\mu}^\nu$ ($\lambda,\mu,\nu$ are strict partitions). The coefficients $g_{\lambda\mu}$ which appear in the decomposition of Schur $Q$-function…

We introduce a Pfaffian formula that extends Schur's $Q$-functions $Q_\lambda$ to be indexed by compositions $\lambda$ with negative parts. This formula makes the Pfaffian construction more consistent with other constructions, such as the…

Let $s_\nu \circ s_\mu$ denote the plethystic product of the Schur functions $s_\nu$ and $s_\mu$. In this article we define an explicit polynomial representation corresponding to $s_\nu \circ s_\mu$ with basis indexed by certain…

We generalize several classical results about Schur functions to the family of cylindric Schur functions. First, we give a combinatorial proof of a Murnaghan--Nakayama formula for expanding cylindric Schur functions in the power-sum basis.…

We give a basis for the space V spanned by the lowest degree part \hat{s}_\lambda of the expansion of the Schur symmetric functions s_\lambda in terms of power sums, where we define the degree of the power sum p_i to be 1. In particular,…

The ring of cyclic quasi-symmetric functions and its non-Escher subring are introduced in this paper. A natural basis consists of fundamental cyclic quasi-symmetric functions; for the non-Escher subring they arise as toric $P$-partition…

In this paper, using the theory of category, we generalize known properties of symmetric polynomials and functions and characterize the multi-indicial symmetric functions. Examples have been given on Schur functions.

It is known that the Grothendieck group of the category of Schur functors is the ring of symmetric functions. This ring has a rich structure, much of which is encapsulated in the fact that it is a "plethory": a monoid in the category of…

We consider the problem of determining when the difference of two ribbon Schur functions is a single Schur function. We fully classify the five infinite families of pairs of ribbon Schur functions whose difference is a single Schur function…