Littlewood–Richardson rules for symmetric skew quasisymmetric Schur functions

Abstract

Abstract. The classical Littlewood-Richardson rule is a rule for computing coefficients in many areas, and comes in many guises. In this paper we prove two Littlewood-Richardson rules for symmetric skew quasisymmetric Schur functions that are analogous to the famed version of the classical Littlewood-Richardson rule involving Yamanouchi words. Furthermore, both our rules contain this classical Littlewood-Richardson rule as a special case. We then apply our rules to combinatorially classify symmetric skew quasisymmetric Schur functions. This answers affirmatively a conjecture of Bessenrodt, Luoto and van Willigenburg.

Noncommutative LR coefficients and crystal reflection operators

“…Once the tableaux in LRT(λ, µ, ν) are computed, it remains to apply an appropriate sequence of crystal reflections (determined by α) followed by an appropriate ρ β to determine all noncommutative LR coefficients C γ αβ . On a related noted, Bessenrodt-Tewari-van Willigenburg [6], in classifying symmetric skew quasisymmetric Schur functions, established Yamanouchi-type rulescalled left and right LR rules therein -for the C γ αβ in the special cases where α is either a partition or a reverse partition, although the proof therein was not uniform. It transpires that these rules are in fact two extremes of our Theorem 3.1.…”

“…It transpires that these rules are in fact two extremes of our Theorem 3.1. Indeed by picking σ to either be the identity or the longest word in S (ν) , we obtain the rules of [6]; see discussion at the end of Section 5 for details.…”

“…We also interpret Theorem 1.1 in terms of chains in Young's lattice indexed by certain frank words and obtain a rule for C γ αβ involving box-adding operators on compositions. In the case where α is either a partition or reverse partition, our interpretations yield the two LR rules in [6]. For an example demonstrating Theorem 1.1, see Subsection 3.2.…”

Quasisymmetric and noncommutative skew Pieri rules

“…The composition of length 4 underlying α is (2,4,3,6), and the partition of length 4 underlying it is (6,4,3,2). They all have size 15.…”

Permuted composition tableaux, 0-Hecke algebra and labeled binary trees

“…Composition tableaux were introduced in [19] to define the basis of quasisymmetric Schur functions for the Hopf algebra of quasisymmetric functions. These functions are analogues of the ubiquitous Schur functions [41], have been studied in substantial detail recently [8,19,20,30,31,45], and have consequently been the genesis of an active new branch of algebraic combinatorics discovering Schur-like bases in quasisymmetric functions [1,5], type B quasisymmetric Schur functions [27,37], quasi-key polynomials [2,42] and quasisymmetric Grothendieck polynomials [35]. Just as Young tableaux play a crucial role in the combinatorics of Schur functions [40,43], composition tableaux are key to understanding the combinatorics of quasisymmetric Schur functions.…”