[Paper Review] Fermionic approach to bilinear expansions of Schur functions in Schur $Q$-functions

J. Harnad, A. Yu. Orlov|arXiv (Cornell University)|Aug 31, 2020

Random Matrices and Applications4 citations

TL;DR

This paper derives a general identity expressing Schur functions as sums of products of Schur $Q$-functions using fermionic Fock space formalism. By leveraging vacuum expectation values (VEVs) of charged and neutral fermionic operators, Wick's theorem, and a factorization identity for anticommuting fermionic pairs, it unifies and generalizes prior special cases of bilinear expansions in Schur $Q$-functions.

ABSTRACT

An identity is derived expressing Schur functions as sums over products of pairs of Schur $Q$-functions, generalizing previously known special cases. This is shown to follow from their representations as vacuum expectation values (VEV's) of products of either charged or neutral fermionic creation and annihilation operators, Wick's theorem and a factorization identity for VEV's of products of two mutually anticommuting sets of neutral fermionic operators.

Motivation & Objective

• To generalize known special cases of bilinear expansions of Schur functions in terms of Schur $Q$-functions.
• To establish a systematic framework for such expansions using fermionic Fock space representations.
• To derive a universal identity connecting Schur functions to products of Schur $Q$-functions through fermionic operator techniques.
• To demonstrate that the identity follows from fundamental properties of vacuum expectation values and fermionic anticommutation relations.

Proposed method

• Represent Schur functions and Schur $Q$-functions as vacuum expectation values (VEVs) of products of charged fermionic creation and annihilation operators.
• Apply Wick's theorem to decompose VEVs of products of fermionic operators into sums of normal-ordered terms.
• Utilize a factorization identity for VEVs of products of two mutually anticommuting sets of neutral fermionic operators to simplify the expressions.
• Leverage the anticommutation relations between fermionic operators to derive the bilinear structure in terms of Schur $Q$-functions.
• Show that the resulting expression generalizes previously known identities by embedding them as special cases.
• Establish the connection between symmetric function theory and fermionic Fock space via operator formalism.

Experimental results

Research questions

• RQ1 How can Schur functions be systematically expressed as bilinear combinations of Schur $Q$-functions?
• RQ2 What underlying fermionic operator structure enables such bilinear expansions?
• RQ3 In what way do vacuum expectation values and Wick's theorem facilitate the derivation of these identities?
• RQ4 How does the factorization identity for anticommuting neutral fermionic operators contribute to the generalization?
• RQ5 What is the relationship between the new identity and previously known special cases of Schur function expansions?

Key findings

• A general identity is derived that expresses any Schur function as a sum of products of pairs of Schur $Q$-functions.
• The identity is shown to follow from the fermionic Fock space representation of Schur and Schur $Q$-functions via vacuum expectation values.
• Wick's theorem provides the combinatorial mechanism to decompose the VEVs into the required bilinear form.
• The factorization identity for mutually anticommuting neutral fermionic operators plays a crucial role in simplifying the VEVs.
• The method unifies and generalizes earlier special-case identities, providing a coherent framework for such expansions.
• The approach demonstrates the power of fermionic operator formalism in symmetric function theory.