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Orthogonality of spin $q$-Whittaker polynomials

Matteo Mucciconi

TL;DR

This work proves that the inhomogeneous spin $q$-Whittaker polynomials $\mathbb{f}_\lambda(\boldsymbol{x}_n|\boldsymbol{a}_{n-1},\boldsymbol{b}_{n-1})$ are self-orthogonal under a torus Sklyanin-type measure and hence form a basis for the space of symmetric polynomials in $n$ variables, with an explicit norm $\mathsf{c}_n(\lambda|\boldsymbol{a}_{n-1},\boldsymbol{b}_{n-1})$. The authors also show that the related family $\mathbb{F}_\lambda(\boldsymbol{x}_n|\boldsymbol{a}_{n},\boldsymbol{b}_{n})$ is a basis, connecting to vanishing-condition characterizations and specializations to spin Grothendieck polynomials, interpolation $q$-Whittaker polynomials, and spin Whittaker functions. Central to the results are inhomogeneous eigenrelations, Cauchy-type identities, and a carefully constructed torus scalar product that generalizes the classical $q$-Whittaker orthogonality. The paper further discusses special cases and formal limits, highlighting links to integrable probabilistic models and potential representation-theoretic interpretations of the orthogonality in the Grothendieck and Whittaker regimes. Overall, the results establish completeness and orthogonality in a rich, multi-parameter setting, providing tools for both algebraic and probabilistic applications in the theory of symmetric functions.

Abstract

The inhomogeneous spin $q$-Whittaker polynomials are a family of symmetric polynomials which generalize the Macdonald polynomials at $t=0$. In this paper we prove that they are orthogonal with respect to a variant of the Sklyanin measure on the $n$ dimensional torus and as a result they form a basis of the space of symmetric polynomials in $n$ variables. Instrumental to the proof are inhomogeneous eigenrelations, which partially generalize those of Macdonald polynomials. We also consider several special cases of the inhomogeneous spin $q$-Whittaker polynomials, which include variants of symmetric Grothendieck polynomials or spin Whittaker functions.

Orthogonality of spin $q$-Whittaker polynomials

TL;DR

This work proves that the inhomogeneous spin -Whittaker polynomials are self-orthogonal under a torus Sklyanin-type measure and hence form a basis for the space of symmetric polynomials in variables, with an explicit norm . The authors also show that the related family is a basis, connecting to vanishing-condition characterizations and specializations to spin Grothendieck polynomials, interpolation -Whittaker polynomials, and spin Whittaker functions. Central to the results are inhomogeneous eigenrelations, Cauchy-type identities, and a carefully constructed torus scalar product that generalizes the classical -Whittaker orthogonality. The paper further discusses special cases and formal limits, highlighting links to integrable probabilistic models and potential representation-theoretic interpretations of the orthogonality in the Grothendieck and Whittaker regimes. Overall, the results establish completeness and orthogonality in a rich, multi-parameter setting, providing tools for both algebraic and probabilistic applications in the theory of symmetric functions.

Abstract

The inhomogeneous spin -Whittaker polynomials are a family of symmetric polynomials which generalize the Macdonald polynomials at . In this paper we prove that they are orthogonal with respect to a variant of the Sklyanin measure on the dimensional torus and as a result they form a basis of the space of symmetric polynomials in variables. Instrumental to the proof are inhomogeneous eigenrelations, which partially generalize those of Macdonald polynomials. We also consider several special cases of the inhomogeneous spin -Whittaker polynomials, which include variants of symmetric Grothendieck polynomials or spin Whittaker functions.

Paper Structure

This paper contains 23 sections, 25 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Context
  3. Inhomogeneous spin $q$-Whittaker polynomials
  4. Main results
  5. Layout of the paper
  6. Notation
  7. Acknowledgements
  8. Symmetric functions
  9. Properties of inhomogeneous spin $q$-Whittaker polynomials
  10. Cauchy identities
  11. Spin Hall-Littlewood Rational Functions
  12. Eigenrelations
  13. Eigenrelations involving $\lambda_n$
  14. Eigenrelations involving $\lambda_1$
  15. Proofs of main theorems
  16. ...and 8 more sections

Key Result

Theorem 1.1

Fix $n \in \mathbb{N}$. For all $\lambda,\mu \in \mathbb{Y}_n$, we have where

Theorems & Definitions (66)

  • Definition 1.1
  • Remark 1.1
  • Remark 1.2
  • Definition 1.2
  • Remark 1.3
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 2.1
  • ...and 56 more