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Neutral-Fermion constructions of factorial $gp$-and $gq$-Functions

Koushik Brahma, Takeshi Ikeda, Shinsuke Iwao, Yi Yang

Abstract

We develop neutral-fermionic constructions for the factorial $gp$- and $gq$-functions introduced by Nakagawa and Naruse, which are respectively dual to the factorial $GQ$- and $GP$-functions of Ikeda and Naruse. In particular, we realize the factorial $GP$-, $GQ$- and $gq$-functions as vacuum expectation values. As applications, we obtain, Jacobi--Trudi type determinantal formulas for the transition coefficients between functions with different equivariant parameters for $gq$ and its dual $GP$, as well as a Pfaffian formula for the factorial $gq$-functions. We further prove a remarkable coincidence among the transition coefficients for parameter changes for $gp$, $gq$, $GQ$, and $GP$. These coefficients admit a description in terms of factorial Grothendieck polynomials of type A.

Neutral-Fermion constructions of factorial $gp$-and $gq$-Functions

Abstract

We develop neutral-fermionic constructions for the factorial - and -functions introduced by Nakagawa and Naruse, which are respectively dual to the factorial - and -functions of Ikeda and Naruse. In particular, we realize the factorial -, - and -functions as vacuum expectation values. As applications, we obtain, Jacobi--Trudi type determinantal formulas for the transition coefficients between functions with different equivariant parameters for and its dual , as well as a Pfaffian formula for the factorial -functions. We further prove a remarkable coincidence among the transition coefficients for parameter changes for , , , and . These coefficients admit a description in terms of factorial Grothendieck polynomials of type A.
Paper Structure (28 sections, 34 theorems, 179 equations)

This paper contains 28 sections, 34 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Factorial $GP,GQ$ and $gp,gq$-functions
  3. Preliminaries on root systems and Weyl groups, and strict partitions
  4. Root systems and Weyl groups
  5. Strict Partitions and Shifted Diagrams
  6. Factorial $GP$- and $GQ$-functions
  7. Notation in the $K$-theoretic Setting
  8. Definition of $GP$ and $GQ$
  9. Vanishing property
  10. Factorization property
  11. Demazure operators
  12. Factorial $gp$- and $gq$-functions
  13. Free-Fermion formalism
  14. Neutral Fermions and Fock spaces
  15. $\beta$ deformed Fermions
  16. ...and 13 more sections

Key Result

Proposition 2.3

Let $\lambda$ and $\mu$ be strict partitions. If $\lambda\not\subseteq\mu$ then $GQ_\lambda(b_\mu|b)=0$, and $GP_\lambda(b_\mu|b)=0$. Moreover, we have $GQ_\lambda(b_\lambda|b)\neq 0$, $GP_\lambda(b_\lambda|b)\neq 0$.

Theorems & Definitions (62)

  • Example 2.1
  • Remark 2.2
  • Proposition 2.3: Vanishing property, IN13
  • Lemma 2.4: Factorization formula for factorial $GP$- and $GQ$-functions
  • proof
  • Theorem 2.5: IN13
  • Theorem 2.6: NN:Uni
  • proof
  • Proposition 3.1: Wick's theorem, DJM, AZ
  • Lemma 3.2: Boson-Fermion correspondence, Iwao2023-2, § 3.4
  • ...and 52 more