Noncommutative Schur functions, switchboards, and Schur positivity

Abstract

We review and further develop a general approach to Schur positivity of symmetric functions based on the machinery of noncommutative Schur functions. This approach unifies ideas of Assaf [1,3], Lam [22], and Greene and the second author [11].

Catalan functions and 𝑘-Schur positivity

“…By a similar argument, (2.6) and (2.8) imply 15) where ω is the involution on Λ defined by ω(e d ) = h d for d ≥ 0. This establishes (2). The properties in (3) determine a unique family of symmetric functions by Theorem 2.7.…”

“…The horizontal dual Pieri rule. We prove this rule (Property (2.5)) using the vertical dual Pieri rule and a trick involving symmetric functions in noncommuting variables, inspired by [9,2].…”

Rules of Three for commutation relations

“…, u N = v N , we obtain the following corollary. Corollary 1.3 ([11], [6]). Let R be a ring, and let u = (u 1 , .…”

“…It was later adapted to study LLT polynomials [13] and k-Schur functions [12]; other variations appeared in [2,11,19]. Further recent work includes the papers [4,5,6], which advance the theory to encompass Lam's work [13] and incorporate ideas of Assaf [1]. One of the main outcomes of this approach is a proof of Haglund's conjecture on 3-column Macdonald polynomials [5].…”

Necessary Conditions for Schur-Maximality

“…where λ and µ are partitions of N and K λµ is the Kostka coefficient. To help explain this phenomenon, in addition to dealing with Schur functions directly, tools have been developed to determine Schur-positivity such as dual equivalence graphs [1,2,6], or the theory of crystal bases that has also been applied as a means to determine Schur-positivity [7]. Despite their rarity, examples of Schur-positive functions arise in a variety of contexts from graph theory and the study of chromatic symmetric functions [10] and chromatic quasisymmetric functions [27] to enumerative combinatorics where sets of partitions determining a quasisymmetric function have been shown to determine a function that is in fact symmetric and Schur-positive [8,11,12].…”