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Quasisymmetric Schur functions

J. Haglund, K. Luoto, S. Mason, S. van Willigenburg

arXiv:0810.2489v2math.CO

TL;DR

The paper seeks quasisymmetric functions that interpolate between Demazure atoms and Schur functions. It defines quasisymmetric Schur functions and derives combinatorial refinements of Schur-function properties, including a Pieri rule; extending the construction to include q remains unresolved.

  • Problem

    The paper asks which Schur-function properties are exhibited by quasisymmetric functions interpolating between Demazure atoms and Schur functions.

    Source: §1 · p. 2

  • Method

    The authors define quasisymmetric Schur functions as a basis of quasisymmetric functions and derive expansions and a Pieri rule for them.

    Source: §1 · p. 3

  • Results

    Quasisymmetric Schur functions lift combinatorial properties indexed by partitions to compositions, including monomial and fundamental expansions and a refined Pieri rule.

    Source: §1 · p. 3

  • Takeaways & Limitations

    The refined Pieri rule defines a poset on compositions that contains Young’s lattice as a subposet.

    Source: §6 · p. 17

    The authors do not see an elegant way to extend the construction to include q because the bijection used in Proposition 7.1 does not apply to fillings with descents just right of basement squares.

    Source: §7.3 · p. 28

Abstract

from arXiv · show

We introduce a new basis for quasisymmetric functions, which arise from a specialization of nonsymmetric Macdonald polynomials to standard bases, also known as Demazure atoms. Our new basis is called the basis of quasisymmetric Schur functions, since the basis elements refine Schur functions in a natural way. We derive expansions for quasisymmetric Schur functions in terms of monomial and fundamental quasisymmetric functions, which give rise to quasisymmetric refinements of Kostka numbers and standard (reverse) tableaux. From here we derive a Pieri rule for quasisymmetric Schur functions that naturally refines the Pieri rule for Schur functions. After surveying combinatorial formulas for Macdonald polynomials, including an expansion of Macdonald polynomials into fundamental quasisymmetric functions, we show how some of our results can be extended to include the $t$ parameter from Hall-Littlewood theory.

1. Introduction

The paper introduces quasisymmetric Schur functions to interpolate between Demazure atoms and Schur functions, lifting familiar properties of symmetric functions to compositions. It derives monomial and fundamental expansions and a Pieri rule, then extends some results to Hall–Littlewood polynomials.

1. Introduction

  • Quasisymmetric Schur functions lift combinatorial properties of symmetric functions indexed by partitions to quasisymmetric functions indexed by compositions.The paper introduces them as an interpolation between Demazure atoms and Schur functions.Source: §1 · pp. 2–3
  • Their monomial expansions yield quasisymmetric Kostka coefficients, refining the Schur-function expression in monomial symmetric functions.The paper also derives a fundamental-quasisymmetric expansion refining the corresponding Schur-function expression.Source: §1 · p. 3
  • The Pieri rule refines multiplication by a row- or column-indexed Schur function to multiplication by a row- or column-indexed quasisymmetric Schur function.The resulting rule uses a poset of compositions and is a new example of the construction studied in.Source: §1 · p. 3
  • Products of two generic quasisymmetric Schur functions can have negative structure constants, unlike the nonnegative constants of the classical Littlewood–Richardson rule.The smallest such example occurs at n = 6; the paper notes a refinement for products of a Schur function and a quasisymmetric Schur function [16].Source: §1 · p. 3
  • The paper extends some results to include the Hall–Littlewood parameter t, defining quasisymmetric functions that decompose Hall–Littlewood polynomials.It contrasts this decomposition with an alternative obtained from a Macdonald-polynomial formula involving fundamental quasisymmetric functions.Source: §1 · p. 3

2. Quasisymmetric and symmetric functions

This section sets up compositions and partitions and introduces quasisymmetric and symmetric functions and their natural bases.

2.1. Compositions and partitions.

  • A weak composition is a list of nonnegative integers summing to n; removing zero parts gives a composition, and reordering positive parts decreasingly gives its associated partition.A composition is a weak composition with all parts positive; a partition is a composition whose parts are weakly decreasing.Source: §2.1 · pp. 3–4
  • Compositions correspond to subsets of [n −1] through partial sums, while coarsening, lexicographic order, and the order ◮ provide relations used in the section.The reversal of a composition lists its parts in reverse order.Source: §2.1 · p. 4

2.2. Quasisymmetric and symmetric function preliminaries.

  • Quasisymmetric functions form a graded algebra, with monomial and fundamental bases indexed by compositions.The symmetric functions form a subalgebra, with Schur functions among their bases.Source: §2.2 · pp. 4–5

3. Schur functions

The section recalls reverse tableaux as a combinatorial model for Schur functions and reviews classical Schur-function expansions and Pieri products.

3.1. Diagrams and reversetableaux.

  • A reversetableau fills a skew diagram with positive integers that weakly decrease across rows and strictly decrease down columns; its weight records the multiplicities of its entries.For a standard reversetableau, i is a descent when i + 1 appears in a higher row; equivalently, i + 1 does not appear strictly left of i.Source: §3.1 · pp. 5–6

3.2. Schur function preliminaries.

  • Schur functions expand in monomial symmetric functions with coefficients counting reversetableaux of specified shape and weight, and in fundamental quasisymmetric functions with terms indexed by standard reversetableaux and their descents.The section gives the example s_(2,1) = m_(2,1) + 2m_(1,1,1) = F_(2,1) + F_(1,2).Source: §3.2 · p. 6
  • The classical Pieri rule multiplies a Schur function by a row- or column-indexed Schur function using horizontal or vertical strips of the specified size.These products are recalled as the rules the paper later refines to the quasisymmetric setting.Source: §3.2 · p. 7

4. Demazure atoms

The section develops composition tableaux and semistandard augmented fillings as combinatorial models connected by a weight-preserving bijection. It then relates these fillings to Demazure atoms and reverse tableaux.

4.1. Compositions and diagrams.

  • Composition tableaux fill composition diagrams with weakly decreasing rows, a strictly increasing leftmost column, and a triple rule; semistandard augmented fillings use a basement and impose non-attacking, no-descent, and inversion-triple conditions.The paper establishes a natural weight-preserving bijection between composition tableaux of shape α and SSAFs whose weak composition collapses to α.Source: §4.1 · pp. 8–9

4.2. Demazure atom preliminaries.

  • A Demazure atom Aγ is defined as a generating function over SSAFs of shape γ, with an equivalent composition-tableau description.Demazure atoms can be used to describe Schur functions.Source: §4.2 · p. 11

4.3. Bijection between reversetableaux and SSAFs.

  • The map ρ−1 converts a reversetableau into an SSAF by placing entries column by column in eligible cells to the right of a basement; removing the basement and zero parts gives a bijection to composition tableaux.Eligible placements must have a filled cell immediately to the left and create no descent.Source: §4.3 · pp. 11–12

5. Quasisymmetric Schur functions

Quasisymmetric Schur functions are defined as composition-indexed analogues of Schur functions, and their expansions show how Schur-function structure carries over to quasisymmetric functions. They form a Z-basis for the algebra of quasisymmetric functions.

5. Quasisymmetric Schur functions

  • Each Schur function decomposes as a sum of quasisymmetric Schur functions indexed by compositions whose associated partition is its shape.This decomposition parallels the expression of monomial symmetric functions using monomial quasisymmetric functions.Source: §5 · p. 13
  • The functions are quasisymmetric, as shown by expressing them in terms of fundamental quasisymmetric functions.The expansion sums over standard reverse tableaux satisfying the stated shape and composition conditions.Source: §5 · p. 13
  • For each composition α, the fundamental function Fα occurs in Sα with coefficient 1.The construction uses the unique standard reverse tableau Tα with β(D(Tα)) = α.Source: §5 · pp. 12–14
  • The quasisymmetric Schur functions form a Z-basis for the algebra of quasisymmetric functions.The basis follows from an upper unitriangular transition matrix to the fundamental quasisymmetric functions.Source: §5 · p. 14

6. Properties of quasisymmetric Schur functions

This section develops properties of quasisymmetric Schur functions through their monomial and fundamental expansions, transition matrices, and a Pieri rule refining the Schur-function rule. It also characterizes when they equal monomial or fundamental quasisymmetric functions and distinguishes the quasisymmetric Schur basis from those in.

6. Properties of quasisymmetric Schur functions

  • The monomial and fundamental expansions have coefficients counted by composition tableaux and standard composition tableaux, respectively.The monomial coefficient Kαβ counts SSAFs of the specified shape and weight, equivalently composition tableaux; the fundamental coefficient dαβ counts standard composition tableaux with the specified descent composition.Source: §6 · pp. 14–15
  • The refined Pieri rule indexes products by horizontal or vertical strips of size n whose row or column operators recover α.For a horizontal strip δ, rowS(δ)(β) = α; for a vertical strip ε, colM(ε)(β) = α. The rule also defines a composition poset that contains Young’s lattice as a subposet.Source: §6 · pp. 16–17

6.1. Proof of the Pieri rule for quasisymmetric Schur functions.

  • The Pieri-rule proof uses reversible Schensted and skyline insertion to match product terms with strip additions, with the vertical-strip case proved analogously.Uninsertion removes the strip cells from right to left in the horizontal case; the row-bumping lemma yields the required ordering.Source: §6.1 · p. 21

6.2. Transition matrices.

  • The transition matrices from quasisymmetric Schur functions to monomial or fundamental functions are upper unitriangular, with tableau-count entries and diagonal entries 1.Their entries count composition tableaux in the monomial case and standard composition tableaux with the specified descent composition in the fundamental case; either matrix can be inverted to expand in the quasisymmetric Schur basis.Source: §6.2 · pp. 21–22
  • Sα equals Mα only for α = (1^f), while Sα equals Fα only for compositions of the stated form.For the fundamental case, α = (m, 1^e1, 2, 1^e2, . . . , 2, 1^f), where m ≠ 1, f ≥ 0, and each ei ≥ 1.Source: §6.2 · p. 22
  • The transition matrix between the quasisymmetric Schur and fundamental bases is the identity for n = 1, 2, and 3.Source: §6.2 · p. 22
  • The quasisymmetric Schur basis differs from the bases appearing in.Source: §6.2 · p. 22

7. Further avenues

The closing sections explore refinements of the Littlewood–Richardson rule, duality for quasisymmetric Schur functions, and Hall–Littlewood decompositions. They identify negative structure constants and the inability of the proposed construction to incorporate q as boundaries, while describing a positive t-dependent decomposition.

7.1. A quasisymmetric refinement of the Littlewood-Richardson rule

  • Products of two generic quasisymmetric Schur functions can have negative structure constants, but mixed Schur–quasisymmetric products admit a refinement with positive integer coefficients.Those coefficients are computed by enumerating composition tableaux subject to a lattice-type condition; the classical Littlewood–Richardson rule is recovered by restricting Demazure characters to Schur polynomials.Source: §7.1 · p. 23

7.2. Skew quasisymmetric functions and duality

  • The dual basis to quasisymmetric Schur functions is addressed using skew quasisymmetric Schur functions in earlier work.Source: §7.2 · p. 24

7.3. Quasisymmetric Hall-Littlewood and Macdonald polynomial decompositions

  • Summing nonsymmetric Hall–Littlewood polynomials over weak compositions collapsing to α defines Lα, which specializes to Sα at t = 0.The polynomials Lα are quasisymmetric, and their sum gives a decomposition of the Hall–Littlewood polynomial into quasisymmetric functions.Source: §7.3 · p. 26
  • At t = 1, Lα equals the monomial quasisymmetric function Mα, while its defining filling formula has coefficients built from nonnegative powers of t and 1 − t.Source: §7.3 · pp. 26–27
  • The alternative q = 0 decomposition of the integral Hall–Littlewood form can have signed coefficient terms, and conversion to Pµ yields coefficients not clearly polynomial in t.The Lα filling formula is described as positive, with each monomial coefficient a sum of terms t^a(1 − t)^b for nonnegative integers a and b.Source: §7.3 · pp. 27–28
  • The authors do not see an elegant way to add q to the Lα construction because the Proposition 7.1 bijection fails for fillings with descents just right of basement squares.Decomposing Lα into fundamental quasisymmetric functions is also left for future research.Source: §7.3 · p. 28
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