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Positive specializations of K-theoretic Schur P- and Q-functions

Eric Marberg

TL;DR

The paper advances the theory of positive specializations by extending Yeliussizov’s Edrei–Thoma-type classification from symmetric Grothendieck functions to the shifted K-theoretic Schur P- and Q-functions, via a robust bialgebra and tableaux framework. It shows that GP^{(\beta)} and GQ^{(\beta)} positive specializations are obtained by restricting G-positive specializations, with explicit product formulas governed by parameters $a$, $\gamma$, and shifted transformations, including the shifted Nazarov-type formulas for one-variable cases. The work also develops single-variable closed forms for border-strip shifted skew shapes and analyzes signed (beta = -1) and unsigned (beta = 1) cases, yielding a comprehensive picture of positivity in the shifted setting. Finally, it connects these algebraic classifications to extreme harmonic functions on shifted (filtered) Young graphs, providing a shifted analogue of the Veršik–Kerov correspondence and extending harmonic analysis on combinatorial lattices. Overall, the results give a complete Edrei–Thoma-type classification for shifted K-theoretic symmetric functions and link them to harmonic-analytic structures on shifted lattices.

Abstract

Yeliussizov has classified the positive specializations of symmetric Grothendieck functions, defined in several different ways, providing a K-theoretic lift of the classical Edrei-Thoma theorem. This note studies the analogous classification problem for Ikeda and Naruse's K-theoretic Schur P- and Q-functions, which are the shifted versions of symmetric Grothendieck functions. Our results extend a shifted variant of the Edrei-Thoma theorem due to Nazarov. We also discuss an application to the problem of determining the extreme harmonic functions on a filtered version of the shifted Young lattice.

Positive specializations of K-theoretic Schur P- and Q-functions

TL;DR

The paper advances the theory of positive specializations by extending Yeliussizov’s Edrei–Thoma-type classification from symmetric Grothendieck functions to the shifted K-theoretic Schur P- and Q-functions, via a robust bialgebra and tableaux framework. It shows that GP^{(\beta)} and GQ^{(\beta)} positive specializations are obtained by restricting G-positive specializations, with explicit product formulas governed by parameters , , and shifted transformations, including the shifted Nazarov-type formulas for one-variable cases. The work also develops single-variable closed forms for border-strip shifted skew shapes and analyzes signed (beta = -1) and unsigned (beta = 1) cases, yielding a comprehensive picture of positivity in the shifted setting. Finally, it connects these algebraic classifications to extreme harmonic functions on shifted (filtered) Young graphs, providing a shifted analogue of the Veršik–Kerov correspondence and extending harmonic analysis on combinatorial lattices. Overall, the results give a complete Edrei–Thoma-type classification for shifted K-theoretic symmetric functions and link them to harmonic-analytic structures on shifted lattices.

Abstract

Yeliussizov has classified the positive specializations of symmetric Grothendieck functions, defined in several different ways, providing a K-theoretic lift of the classical Edrei-Thoma theorem. This note studies the analogous classification problem for Ikeda and Naruse's K-theoretic Schur P- and Q-functions, which are the shifted versions of symmetric Grothendieck functions. Our results extend a shifted variant of the Edrei-Thoma theorem due to Nazarov. We also discuss an application to the problem of determining the extreme harmonic functions on a filtered version of the shifted Young lattice.
Paper Structure (15 sections, 31 theorems, 108 equations)

This paper contains 15 sections, 31 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Positive specializations
  3. Shifted variants
  4. Applications to harmonic functions
  5. Preliminaries
  6. Bialgebra specializations
  7. Symmetric functions
  8. Shifted symmetric functions
  9. Positive K-theoretic specializations
  10. Set-valued tableaux
  11. Unbounded symmetric functions
  12. Unsigned specializations
  13. Single-variable formulas
  14. Signed specializations
  15. Harmonic functions

Key Result

Theorem 1.1

An algebra morphism $\rho : \Gamma \to \mathbb{R}$ is a $G$-positive specialization of $\Gamma$ if and only if for some choice of $a$, $b$, and $\gamma$ satisfying $\max(b) < 1 \leq C =1+\rho(G_1)<\infty$ one has

Theorems & Definitions (66)

  • Theorem 1.1: Yel20
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.5: Edei--Thoma; see Yel20
  • Theorem 2.6: Nazarov Nazarov
  • Definition 3.1
  • Example 3.2
  • ...and 56 more