Generating functions of dual $K$-theoretic $P$- and $Q$-functions and boson-fermion correspondence

Technical Summary: Generating Functions of Dual K-Theoretic P- and Q-Functions and Boson-Fermion Correspondence

Problem Statement
The paper addresses the algebraic characterization of K-theoretic Schur P- and Q-functions ($G P_\lambda$ and $G Q_\lambda$) and their duals ($g p_\lambda$ and $g q_\lambda$), introduced by Ikeda and Naruse. These functions represent Schubert classes in the K-theory of maximal isotropic symplectic and orthogonal Grassmannians. While the K-theoretic Q-functions ($G Q_\lambda$) had previously been characterized using $\beta$-deformed neutral fermion operators (in a prior work by the author), a complete fermionic presentation for the K-theoretic P-functions ($G P_\lambda$) and, crucially, their dual functions ($g p_\lambda$ and $g q_\lambda$) remained an open question. Specifically, Nakagawa-Naruse had conjectured generating functions for these dual functions, but a rigorous derivation via the boson-fermion correspondence was lacking.

Methodology
The author employs the framework of neutral fermion operators within a Fock space formalism, extending the standard theory to include a deformation parameter $\beta$. The core methodological innovations include:

1. Introduction of Four Families of $\beta$-Deformed Operators:
   The paper defines four distinct families of neutral fermion operators depending on $\beta$: $\phi^{(\beta)}_n$, $\phi^{[\beta]}_n$, $\Phi^{(\beta)}_n$, and $\Phi^{[\beta]}_n$.

   • $\phi^{(\beta)}_n$ and $\Phi^{(\beta)}_n$ are constructed to generate the K-theoretic functions $G Q_\lambda$ and $G P_\lambda$.
   • $\phi^{[\beta]}_n$ and $\Phi^{[\beta]}_n$ are constructed to generate the dual functions $g q_\lambda$ and $g p_\lambda$.
     These operators are defined as infinite linear combinations of standard neutral fermion operators $\phi_n$, involving specific shifts and scaling by $\beta$.

2. Definition of Dual Pairs and Commutation Relations:
   The author establishes a duality relation between the operator families. Specifically, the anti-commutator relations are derived such that $[(\Phi^{(\beta)}_m)^*, \phi^{[\beta]}_n]_+ = [(\Phi^{[\beta]}_m)^*, \phi^{(\beta)}_n]_+ = \delta_{m,n}$. This serves as the $\beta$-deformed analog of the standard duality relation in the undeformed case.

3. $\beta$-Deformed Boson-Fermion Correspondence:
   Two linear maps, $\Omega$ and $\chi$, are introduced as $\beta$-deformed boson-fermion correspondences.

   • $\Omega$ maps vectors in the fermion Fock space to the completed ring of symmetric functions $\widehat{\Lambda}$, corresponding to the K-theoretic functions.
   • $\chi$ maps vectors to a subalgebra $g\Gamma$, corresponding to the dual functions.
     These maps are defined via vacuum expectation values involving $\beta$-deformed Hamiltonian operators $H^{(\beta)}(x)$ and $H^{[\beta]}(x)$.

4. Derivation of Generating Functions:
   By exploiting commutation relations among the $\beta$-deformed vertex operators and current operators, the author derives the K-theoretic Cauchy kernel of Nakagawa-Naruse. This kernel connects the vacuum expectation values of the fermionic operators to the product formula $\prod_{i,j} \frac{1 - \bar{x}_i y_j}{1 - x_i y_j}$ (where $\bar{x}_i$ is a $\beta$-shifted variable).

Key Contributions and Results
The primary results of the paper are the explicit realization of the four K-theoretic functions as vacuum expectation values of specific fermionic operators and the derivation of generating functions for the dual functions.

1. Fermionic Presentations:
   Theorem 1.1 establishes that for any strict partition $\lambda$, there exist four vectors $|\lambda\rangle_Q, |\lambda\rangle_P, |\lambda\rangle_q, |\lambda\rangle_p$ in the fermion Fock space such that:

   • $G Q_\lambda = \Omega(|\lambda\rangle_Q)$
   • $G P_\lambda = \Omega(|\lambda\rangle_P)$
   • $g q_\lambda = \chi(|\lambda\rangle_q)$
   • $g p_\lambda = \chi(|\lambda\rangle_p)$
     The vectors are constructed using the respective $\beta$-deformed fermion operators defined in the text.

2. Generating Functions for Dual Functions:
   Using the orthonormality of the constructed vectors and the properties of the $\beta$-deformed boson-fermion correspondence, the paper derives explicit generating functions for the dual K-theoretic functions $g q_\lambda$ and $g p_\lambda$. This confirms the conjecture made by Nakagawa-Naruse regarding the form of these generating functions.

3. K-Theoretic Cauchy Identity:
   The paper demonstrates that the commutation relations of the $\beta$-deformed operators naturally lead to the K-theoretic Cauchy identity:
   $$\sum_{\lambda: \text{strict}} G Q_\lambda(x) g p_\lambda(y) = \sum_{\lambda: \text{strict}} G P_\lambda(x) g q_\lambda(y) = \prod_{i,j} \frac{1 - \bar{x}_i y_j}{1 - x_i y_j}$$
   where $\bar{x}_i = \frac{-x_i}{1 + \beta x_i}$.

Significance
The paper provides a unified algebraic framework for K-theoretic symmetric functions of types B, C, and D. By extending the boson-fermion correspondence to include $\beta$-deformed operators, the author successfully realizes the K-theoretic P-functions and their duals in a manner analogous to the previously known Q-functions. This presentation not only validates the conjectured generating functions for the dual functions but also offers a systematic method for studying the algebraic properties of these functions through the lens of neutral fermion operators. The work generalizes previous fermionic descriptions of type A K-theoretic functions (such as stable Grothendieck polynomials) to the orthogonal and symplectic cases.