Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the Boson-Fermion Correspondence for Factorial Schur Functions

Daniel Bump, Andrew Hardt, Travis Scrimshaw

TL;DR

The work provides a fully algebraic realization of Molev's deformed boson-fermion correspondence in the $\boldsymbol{\beta}=0$ setting, removing analytic assumptions by working over $\mathbb{Z}[\boldsymbol{\alpha}]$ and using Laurent-series methods. It builds a representation-theoretic framework via a completion of $\mathfrak{gl}_{\infty}$ and a central extension to a Heisenberg algebra, identifies the fermionic basis with Molev's double supersymmetric Schur functions and their duals, and derives Jacobi–Trudi-type constructions, raising-operator formulas, and skew-Pieri rules for these objects. The results give finite-sum, purely algebraic descriptions of factorial/double Schur theory and connect to lattice-model transfer matrices through deformed half-vertex operators, offering new tools for representation theory and algebraic combinatorics of supersymmetric Schur functions.

Abstract

We give an algebraic (non-analytic) proof of the deformed boson-fermion Fock space construction of Molev's double supersymmetric Schur functions, among other results, from our previous paper. In other words, we make no assumptions on the variables and parameters. By specializing to a finite number of variables and shifting parameters, we recover the factorial Schur functions. Furthermore, we realize the bosonic construction through a representation of a completion of the infinite rank general linear Lie algebra.

On the Boson-Fermion Correspondence for Factorial Schur Functions

TL;DR

The work provides a fully algebraic realization of Molev's deformed boson-fermion correspondence in the setting, removing analytic assumptions by working over and using Laurent-series methods. It builds a representation-theoretic framework via a completion of and a central extension to a Heisenberg algebra, identifies the fermionic basis with Molev's double supersymmetric Schur functions and their duals, and derives Jacobi–Trudi-type constructions, raising-operator formulas, and skew-Pieri rules for these objects. The results give finite-sum, purely algebraic descriptions of factorial/double Schur theory and connect to lattice-model transfer matrices through deformed half-vertex operators, offering new tools for representation theory and algebraic combinatorics of supersymmetric Schur functions.

Abstract

We give an algebraic (non-analytic) proof of the deformed boson-fermion Fock space construction of Molev's double supersymmetric Schur functions, among other results, from our previous paper. In other words, we make no assumptions on the variables and parameters. By specializing to a finite number of variables and shifting parameters, we recover the factorial Schur functions. Furthermore, we realize the bosonic construction through a representation of a completion of the infinite rank general linear Lie algebra.

Paper Structure

This paper contains 12 sections, 24 theorems, 104 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Shifted powers
  4. Symmetric function identities
  5. Lie algebra representations
  6. Near diagonal infinite matrices
  7. Current operators and near triangular infinite matrices
  8. Central extensions
  9. Double supersymmetric functions
  10. The Boson-Fermion Correspondence
  11. Raising operator formulas
  12. Skew-Pieri rule

Key Result

Theorem 1.1

Let $\lambda$ be a partition. There exists an $\boldsymbol{\alpha}$ deformation of the classical boson-fermion correspondence such that the image of the basis vector $\lvert \lambda \rangle$ of fermionic Fock space is Molev's double supersymmetric Schur function MolevFactorialSupersymmetricMolev09 o

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 2.1
  • Example 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5: BHS
  • Proposition 2.6
  • ...and 45 more