Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generating functions of $W_{1+\infty}$ action on symmetric functions

Caleb Fernelius, Natasha Rozhkovskaya

TL;DR

This work develops a formal-distributions framework to describe the action of the ${W_{1+ abla}}$ algebra and its B-type analogue on symmetric functions, notably Schur and Schur $Q$-functions. It provides compact, derivation-free generating formulas for these actions, reveals a self-duality between the $x$- and $u$-variable components, and places the algebras in a conformal-Lie context with explicit Heisenberg and Virasoro substructures. The authors realize the algebras via charged and neutral free fermions, connect to the boson–fermion correspondence, and derive explicit action formulas on Schur and Schur $Q$-functions, including dual Cauchy-type identities. The results unify and extend known formulas (e.g., LY3-type actions) and illuminate new multiplicative actions on symmetric-function bases with potential applications to KP/BKP tau-functions and related enumerative geometry contexts.

Abstract

We describe the action of the infinite-dimensional Lie algebra $W_{1+\infty}$ and its B-type analogue on Schur and Schur Q-functions, respectively, using formal distributions framework. We observe an interesting self-duality property possessed by these compact formulas.

Generating functions of $W_{1+\infty}$ action on symmetric functions

TL;DR

This work develops a formal-distributions framework to describe the action of the algebra and its B-type analogue on symmetric functions, notably Schur and Schur -functions. It provides compact, derivation-free generating formulas for these actions, reveals a self-duality between the - and -variable components, and places the algebras in a conformal-Lie context with explicit Heisenberg and Virasoro substructures. The authors realize the algebras via charged and neutral free fermions, connect to the boson–fermion correspondence, and derive explicit action formulas on Schur and Schur -functions, including dual Cauchy-type identities. The results unify and extend known formulas (e.g., LY3-type actions) and illuminate new multiplicative actions on symmetric-function bases with potential applications to KP/BKP tau-functions and related enumerative geometry contexts.

Abstract

We describe the action of the infinite-dimensional Lie algebra and its B-type analogue on Schur and Schur Q-functions, respectively, using formal distributions framework. We observe an interesting self-duality property possessed by these compact formulas.

Paper Structure

This paper contains 35 sections, 24 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Lie subalgebra $\mathcal{W}_{1+\infty}$ of $\hat{a}_\infty$
  3. Lie algebra $\hat{a}_\infty$
  4. Formal distributions
  5. Infinite-dimensional Lie algebra $\mathcal{W}_{1+\infty}$
  6. $\mathcal{W}_{1+\infty}$ as the central extension of the algebra of differential operators
  7. Definition of conformal Lie algebra
  8. Conformal Lie algebra structure of $\mathcal{W}_{1+\infty}$
  9. Heisenberg and Virasoro algebras
  10. Presentation of $\hat{a}_\infty$ and $\mathcal{W}_{1+\infty}$ through fermions
  11. Clifford algebra of charged free fermions
  12. Presentation through charged free fermions
  13. Review of properties of symmetric functions
  14. On the boson-ferimon correspondance
  15. Symmetric functions
  16. ...and 20 more sections

Key Result

Lemma 2.1

and In particular,

Theorems & Definitions (59)

  • Remark 2.1
  • Lemma 2.1
  • proof
  • Proposition 2.1
  • proof
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.2
  • proof
  • ...and 49 more