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Pfaffian Formulation of Schur's $Q$-functions

John Graf, Naihuan Jing

TL;DR

This work resolves inconsistencies in extending Schur's Q-functions to compositions with negative parts by introducing a Pfaffian framework with a skew-symmetric matrix M(lambda) that yields well-defined Q_lambda for all lambda. It develops a vertex-operator style identity, provides a precise decomposition of Q_lambda into skew components, and proves key equalities such as Q_{lambda0}=Q_lambda and Q_{lambda/0}=Q_lambda, enabling algebraic proofs of numerous Q-function identities. The main result expresses Q_{p lambda} in terms of Q_lambda and skew functions Q_{lambda/(r)}, and the authors connect the coefficients arising in these decompositions to A_{n-1} root-system data, including staircase partitions and Kostant-type decompositions. Collectively, the paper offers a unified Pfaffian-based approach to Schur's Q-functions, yielding new identities and a versatile toolkit for manipulations with skew and negative parts, with potential implications for symmetric-function theory and related algebraic structures.

Abstract

We introduce a Pfaffian formula that extends Schur's $Q$-functions $Q_λ$ to be indexed by compositions $λ$ with negative parts. This formula makes the Pfaffian construction more consistent with other constructions, such as the Young tableau and Vertex Operator constructions. With this construction, we develop a proof technique involving decomposing $Q_λ$ into sums indexed by partitions with removed parts. Consequently, we are able to prove several identities of Schur's $Q$-functions using only simple algebraic methods.

Pfaffian Formulation of Schur's $Q$-functions

TL;DR

This work resolves inconsistencies in extending Schur's Q-functions to compositions with negative parts by introducing a Pfaffian framework with a skew-symmetric matrix M(lambda) that yields well-defined Q_lambda for all lambda. It develops a vertex-operator style identity, provides a precise decomposition of Q_lambda into skew components, and proves key equalities such as Q_{lambda0}=Q_lambda and Q_{lambda/0}=Q_lambda, enabling algebraic proofs of numerous Q-function identities. The main result expresses Q_{p lambda} in terms of Q_lambda and skew functions Q_{lambda/(r)}, and the authors connect the coefficients arising in these decompositions to A_{n-1} root-system data, including staircase partitions and Kostant-type decompositions. Collectively, the paper offers a unified Pfaffian-based approach to Schur's Q-functions, yielding new identities and a versatile toolkit for manipulations with skew and negative parts, with potential implications for symmetric-function theory and related algebraic structures.

Abstract

We introduce a Pfaffian formula that extends Schur's -functions to be indexed by compositions with negative parts. This formula makes the Pfaffian construction more consistent with other constructions, such as the Young tableau and Vertex Operator constructions. With this construction, we develop a proof technique involving decomposing into sums indexed by partitions with removed parts. Consequently, we are able to prove several identities of Schur's -functions using only simple algebraic methods.
Paper Structure (20 sections, 23 theorems, 110 equations)

This paper contains 20 sections, 23 theorems, 110 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compositions and Partitions
  4. Pfaffians
  5. Schur's $Q$-functions
  6. Fundamental Properties of $Q_\lambda$
  7. Accounting for the Disparity of $\ell(\lambda)$
  8. Reordering the Parts of $\lambda$
  9. Removing Negative Parts
  10. Vertex Operator Identity
  11. Inner Products and Adjoints
  12. The Main Identity
  13. Expanding $Q_\lambda$ Using Skew Functions
  14. Sum of $Q_{\lambda/(r)}$
  15. Sum of $Q_{\lambda/(\lambda_k)}$
  16. ...and 5 more sections

Key Result

Proposition 3.1

Let $\lambda\in\mathbb{Z}^n$ be a composition, then

Theorems & Definitions (45)

  • Proposition 3.1
  • proof
  • Proposition 3.2
  • proof
  • Corollary 3.3
  • proof
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 35 more