Free fermions and Schur expansions of multi-Schur functions

TL;DR

This paper introduces a new free-fermionic framework for multi-Schur functions, enabling explicit expansions and duality conditions, thus advancing the algebraic understanding of these generalized symmetric functions.

Contribution

It provides a novel free-fermionic presentation of multi-Schur functions and a straightforward method for their expansion in refined dual Grothendieck polynomials.

Findings

Constructed linear bases of fermionic Fock space indexed by multi-Schur data

Established correspondence between bases and multi-Schur functions via boson-fermion correspondence

Provided conditions for Hall-duality of multi-Schur functions

Abstract

Multi-Schur functions are symmetric functions that generalize the supersymmetric Schur functions, the flagged Schur functions, and the refined dual Grothendieck functions, which have been intensively studied by Lascoux. In this paper, we give a new free-fermionic presentation of them. The multi-Schur functions are indexed by a partition and two ``tuples of tuples'' of indeterminates. We construct a family of linear bases of the fermionic Fock space that are indexed by such data and prove that they correspond to the multi-Schur functions through the boson-fermion correspondence. By focusing on some special bases, which we call refined bases, we give a straightforward method of expanding a multi-Schur function in the refined dual Grothendieck polynomials. We also present a sufficient condition for a multi-Schur function to have its Hall-dual function in the completed ring of symmetric functions.