# Neutral-fermionic presentation of the $K$-theoretic $Q$-function

## TL;DR

This paper introduces a new neutral-fermionic framework for $K$-theoretic $Q$-functions, simplifying their description and proofs of Pfaffian formulas, and constructs dual functions with a compatible bilinear form.

## Contribution

It provides a novel neutral-fermionic presentation of $K$-theoretic $Q$-functions and establishes dual functions with a compatible bilinear form.

## Key findings

- Simplified Pfaffian formulas for $K$-theoretic $Q$-functions.
- Construction of a dual space with a compatible bilinear form.
- Introduction of new dual $K$-theoretic $Q$-functions with Pfaffian formulas.

## Abstract

We show a new neutral-fermionic presentation of Ikeda-Naruse's $K$-theoretic $Q$-functions, which represent a Schubert class in the $K$-theory of coherent sheaves on the Lagrangian Grassmannian. Our presentation provides a simple description and yields straightforward proof of two types of Pfaffian formulas for them. We present a dual space of $G\Gamma$, the vector space generated by all $K$-theoretic $Q$-functions, by constructing a non-degenerate bilinear form that is compatible with the neutral fermionic presentation. We give a new family of dual $K$-theoretic $Q$-functions, their neutral-fermionic presentations, and Pfaffian formulas.