Related papers: Noncommutative Schur functions for posets

In the Hopf algebra of symmetric functions, Sym, the basis of Schur functions is distinguished since every Schur function is isomorphic to an irreducible character of a symmetric group under the Frobenius characteristic map. In this note we…

The $e$-positivity conjecture and the $e$-unimodality conjecture of chromatic quasisymmetric functions are proved for some classes of natural unit interval orders. Recently, J. Shareshian and M. Wachs introduced chromatic quasisymmetric…

Chen proposed a conjecture on the log-concavity of the generating function for the symmetric group with respect to the length of longest increasing subsequences of permutations. Motivated by Chen's log-concavity conjecture, B\'{o}na,…

Plethysm is a fundamental operation in symmetric function theory, derived directly from its connection with representation theory. However, it does not admit a simple combinatorial interpretation, and finding coefficients of Schur function…

We introduce non-commutative analogues of $k$-Schur functions of Lapointe-Lascoux and Morse. We give an explicit formulas for the expansions of non-commutive functions with one and two parameters in terms of these new functions. These…

We introduce a quasisymmetric generalization of Berele and Regev's hook Schur functions and prove that these new quasisymmetric hook Schur functions decompose the hook Schur functions in a natural way. We examine the combinatorics of the…

By establishing relations between operators on compositions, we show that the posets of compositions arising from the right and left Pieri rules for noncommutative Schur functions can each be endowed with both the structure of dual graded…

We define a new family of symmetric functions which are affine analogues of Stanley symmetric functions. We establish basic properties of these functions including symmetry, dominance and conjugation. We conjecture certain positivity…

Normal crystals (also known as Stembridge crystals) are commonly used to establish the Schur positivity of symmetric functions, as their characters are sums of Schur polynomials. In this paper, we develop a combinatorial framework for a…

We prove that the set of matchings with a fixed number of unmatched vertices is Schur-positive with respect to the set of short chords. Two proofs are presented. The first proof applies a new combinatorial criterion for Schur-positivity,…

The question of classifying when two skew Schur functions are equal is a substantial open problem, which remains unsolved for over a century. In 2022, Aliniaeifard, Li and van Willigenburg introduced skew Schur functions in noncommuting…

We prove a general inclusion-exclusion relation for the extended chromatic symmetric function of a weighted graph, which specializes to (extended) $k$-deletion, and we give two methods to obtain numerous new bases from weighted graphs for…

Let $P$ be a poset, $inc(P)$ its incomparability graph, and $X_{inc(P)}$ the corresponding chromatic symmetric function, as defined by Stanley in {\em Adv. Math.}, {\bf 111} (1995) pp.~166--194. Certain conditions on $P$ imply that the…

In work with A. Yong, the author introduced genomic tableaux to prove the first positive combinatorial rule for the Littlewood-Richardson coefficients in torus-equivariant $K$-theory of Grassmannians. We then studied the genomic Schur…

In 1995 Stanley conjectured that the chromatic symmetric functions of the graphs $P_{d,2}$, which we call triangular ladders, were $e$-positive. In this paper we confirm this conjecture, which is also an unsolved case of the celebrated…

We investigate the $e$-positivity and Schur positivity of the chromatic symmetric functions of some spider graphs with three legs. We obtain the positivity classification of all broom graphs and that of most double broom graphs. The methods…

In this note we classify when a skew Schur function is a positive linear combination of power sum symmetric functions. We then use this to determine precisely when any scalar multiple of a skew Schur function is the chromatic symmetric…

Skewing operators play a central role in the symmetric function theory because of the importance of the product structure of the symmetric function space. The theory of noncommutative symmetric functions is a useful tool for studying…

H.Choi-D.Kim-S.J.Lee and S.J.Lee introduced a new kind of tableaux, semistandard oscillating tableaux (SSOT), around 2024 in the context of Lusztig $q$-weight multiplicities, KR crystals and King tableaux. In this paper, we study generating…

We study a linear map on symmetric functions that ``divides'' a partition by a positive integer $k$, sending a Schur function indexed by a partition of $kn$ to a symmetric function indexed by partitions of $n$. We determine its Schur…