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A Combinatorial Perspective on the Noncommutative Symmetric Functions

Angela Hicks, Robert McCloskey

TL;DR

This paper provides a self-contained, elementary treatment of noncommutative symmetric functions NSym by realizing NSym bases concretely in infinitely many noncommuting variables and avoiding quasideterminants. It illuminates NSym’s connections to Λ and QSym through explicit basis descriptions, dualities, and generating-series relations, and recasts change-of-basis phenomena in NSym/QSym as both matrix-factorization analogues of Macdonald’s framework and as combinatorial statistics on brick tabloids extended to walls. By developing noncommutative analogues of the classical bases (ribbon, elementary, complete, and power sums) and their interrelations, the work provides explicit realizations and a unified combinatorial interpretation of many transition matrices and basis expansions. The results offer a bridge between algebraic structures in Hopf algebras of symmetric-type functions and combinatorial models (bricks and walls) that generalize well-known interpretations from Λ, with potential implications for representation theory via 0-Hecke algebras and related combinatorial Hopf algebras.

Abstract

The noncommutative symmetric functions $\textbf{NSym}$ were first defined abstractly by Gelfand et al. in 1995 as the free associative algebra generated by noncommuting indeterminants $\{\boldsymbol{e}_n\}_{n\in \mathbb{N}}$ that were taken as a noncommutative analogue of the elementary symmetric functions. The resulting space was thus a variation on the traditional symmetric functions $Λ$. Giving noncommutative analogues of generating function relations for other bases of $Λ$ allowed Gelfand et al. to define additional bases of $\textbf{NSym}$ and then determine change-of-basis formulas using quasideterminants. In this paper, we aim for a self-contained exposition that expresses these bases concretely as functions in infinitely many noncommuting variables and avoids quasideterminants. Additionally, we look at the noncommutative analogues of two different interpretations of change-of-basis in $Λ$: both as a product of a minimal number of matrices, mimicking Macdonald's exposition of $Λ$ in Symmetric Functions and Hall Polynomials, and as statistics on brick tabloids, as in work by Eğecioğlu and Remmel, 1990.

A Combinatorial Perspective on the Noncommutative Symmetric Functions

TL;DR

This paper provides a self-contained, elementary treatment of noncommutative symmetric functions NSym by realizing NSym bases concretely in infinitely many noncommuting variables and avoiding quasideterminants. It illuminates NSym’s connections to Λ and QSym through explicit basis descriptions, dualities, and generating-series relations, and recasts change-of-basis phenomena in NSym/QSym as both matrix-factorization analogues of Macdonald’s framework and as combinatorial statistics on brick tabloids extended to walls. By developing noncommutative analogues of the classical bases (ribbon, elementary, complete, and power sums) and their interrelations, the work provides explicit realizations and a unified combinatorial interpretation of many transition matrices and basis expansions. The results offer a bridge between algebraic structures in Hopf algebras of symmetric-type functions and combinatorial models (bricks and walls) that generalize well-known interpretations from Λ, with potential implications for representation theory via 0-Hecke algebras and related combinatorial Hopf algebras.

Abstract

The noncommutative symmetric functions were first defined abstractly by Gelfand et al. in 1995 as the free associative algebra generated by noncommuting indeterminants that were taken as a noncommutative analogue of the elementary symmetric functions. The resulting space was thus a variation on the traditional symmetric functions . Giving noncommutative analogues of generating function relations for other bases of allowed Gelfand et al. to define additional bases of and then determine change-of-basis formulas using quasideterminants. In this paper, we aim for a self-contained exposition that expresses these bases concretely as functions in infinitely many noncommuting variables and avoids quasideterminants. Additionally, we look at the noncommutative analogues of two different interpretations of change-of-basis in : both as a product of a minimal number of matrices, mimicking Macdonald's exposition of in Symmetric Functions and Hall Polynomials, and as statistics on brick tabloids, as in work by Eğecioğlu and Remmel, 1990.
Paper Structure (14 sections, 34 theorems, 138 equations, 4 figures)

This paper contains 14 sections, 34 theorems, 138 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Partitions and Compositions
  4. Three Rings
  5. Bases of Sym and QSym
  6. Well-Known Bases of Sym
  7. Involutions and the Hall inner product on Sym
  8. Well-Known Bases of QSym
  9. Bases for NSym
  10. Relations between transition matrices
  11. Change-of-Basis Expressions in NSym and QSym
  12. Combinatorial Models for Change of Basis
  13. Combinatorial Models in Sym
  14. Brick Walls and Change of Basis in QSym and NSym.

Key Result

Theorem 1.1

Every symmetric function can be written uniquely as a polynomial in the elementary symmetric functions $\{e_n\}_{n\in \mathbb{N}}$.

Figures (4)

  • Figure 1: Images of $\omega$ and Self-Duality of $\Lambda$
  • Figure 2: Automorphisms and Duality of $\textnormal{QSym}$ and $\textbf{NSym}$
  • Figure 3: Matrix Change-of-Basis Expressions in $\Lambda$
  • Figure 4: Matrix Change-of-Basis Expressions in $\textnormal{QSym}$ and $\textbf{NSym}$

Theorems & Definitions (121)

  • Theorem 1.1
  • Definition 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Example 2.6
  • Definition 2.7
  • Example 2.8
  • ...and 111 more