# $\pi$-type Fermions and $\pi$-type KP hierarchy

## TL;DR

This paper introduces $$-type Fermions and Boson-Fermion correspondence, leading to a new $$-type KP hierarchy and associated symmetric functions, expanding the mathematical framework of integrable systems.

## Contribution

It develops the concept of $$-type Fermions and generalizes the Boson-Fermion correspondence, resulting in new $$-type symmetric functions and a generalized KP hierarchy.

## Key findings

- Construction of $$-type Fermions
- Definition of $$-type Boson-Fermion correspondence
- Formulation of $$-type KP hierarchy and tau functions

## Abstract

In this paper, we firstly construct $\pi$-type Fermions. According to these, we define $\pi$-type Boson-Fermion correspondence which is a generalization of the classical Boson-Fermion correspondence. We can obtain $\pi$-type symmetric functions $S_\lambda^\pi$ from the $\pi$-type Boson-Fermion correspondence, analogously to the way we get the Schur functions $S_\lambda$ from the classical Boson-Fermion correspondence (which is the same thing as the Jacobi-Trudi formula). Then as a generalization of KP hierarchy, we construct the $\pi$-type KP hierarchy and obtain its tau functions.