Related papers: The genomic Schur function is fundamental-positive
Let $\lambda$ be a partition of a positive integer $n$. The genomic Schur function $U_\lambda$ was introduced by Pechenik--Yong in the context of the $K$-theory of Grassmannians. Recently, Pechenik provided a positive combinatorial formula…
Let $\Lambda$ be the space of symmetric functions and $V_k$ be the subspace spanned by the modified Schur functions $\{S_\lambda[X/(1-t)]\}_{\lambda_1\leq k}$. We introduce a new family of symmetric polynomials,…
The product $s_\mu s_\nu$ of two Schur functions is one of the most famous examples of a Schur-positive function, i.e. a symmetric function which, when written as a linear combination of Schur functions, has all positive coefficients. We…
Determining if a symmetric function is Schur-positive is a prevalent and, in general, a notoriously difficult problem. In this paper we study the Schur-positivity of a family of symmetric functions. Given a partition \lambda, we denote by…
We introduce and study a generalization $s_{(\mu|\lambda)}$ of the Schur functions called the almost symmetric Schur functions. These functions simultaneously generalize the finite variable key polynomials and the infinite variable Schur…
We study the class $\mathcal C$ of symmetric functions whose coefficients in the Schur basis can be described by generating functions for sets of tableaux with fixed shape. Included in this class are the Hall-Littlewood polynomials,…
We obtain the Schur positivity of spider graphs of the forms $S(a,2,1)$ and $S(a,4,1)$, which are considered to have the simpliest structures for which the Schur positivity was unknown. The proof outline has four steps. First, we find…
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…
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 introduce a new combinatorial object, semistandard increasing decomposition tableau and study its relation to a semistandard decomposition tableau introduced by Kra\'skiewicz and developed by Lam and Serrano. We also introduce…
The theory of noncommutative Schur functions can be used to obtain positive combinatorial formulae for the Schur expansion of various classes of symmetric functions, as shown by Fomin and Greene. We develop a theory of noncommutative super…
Cylindric skew Schur functions, which are a generalisation of skew Schur functions, arise naturally in the study of P-partitions. Also, recent work of A. Postnikov shows they have a strong connection with a problem of considerable current…
We define a number of related combinatorial objects, each of which possesses a surprising symmetry. We include several applications such as a combinatorial explanation for certain fixed points of the involution $\omega$ on the ring of…
We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian…
In the prequel to this paper, we showed how results of Mason involving a new combinatorial formula for polynomials that are now known as Demazure atoms (characters of quotients of Demazure modules, called standard bases by Lascoux and…
We study the space, $R_m$, of $m$-symmetric functions consisting of polynomials that are symmetric in the variables $x_{m+1},x_{m+2},x_{m+3},\dots$ but have no special symmetry in the variables $x_1,\dots,x_m$. We obtain $m$-symmetric…
Given an element in a finite-dimensional real vector space, $V$, that is a nonnegative linear combination of basis vectors for some basis $B$, we compute the probability that it is furthermore a nonnegative linear combination of basis…
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…
Let $\mathbb{N}$ denote the set of non-negative integers. Haglund, Wilson, and the second author have conjectured that the coefficient of any Schur function $s_\lambda[X]$ in $\Delta_{e_k} e_n[X]$ is a polynomial in $\mathbb{N}[q,t]$. We…
We present positivity conjectures for the Schur expansion of Jack symmetric functions in two bases given by binomial coefficients. Partial results suggest that there are rich combinatorics to be found in these bases, including Eulerian…