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Totally positive Toeplitz matrices: classical and modern

Konstanze Rietsch

TL;DR

The paper studies infinite totally positive Toeplitz matrices as limits of finite ones, anchored by Edrei's generating-function factorization, Thoma's parametrization of $S_ fty$ characters, and Vershik–Kerov's asymptotics. It develops two parallel asymptotic descriptions of Schoenberg parameters: one via quantum cohomology and Peterson's isomorphism, connecting to Schubert classes and Chern roots, and another through tropical geometry and tropical versions of Edrei's theorem, relating tropical Schoenberg parameters to Lusztig weights. The work unifies representation theory, quantum cohomology, and tropical methods to describe limiting spectral data and Schubert-theoretic behavior of Toeplitz matrices. It also provides a tropical analogue of the Edrei theory, including two versions of Edrei-type theorems and a tropical VK-type asymptotics, highlighting deep connections between algebraic combinatorics, geometry, and the canonical basis. Overall, the results give a coherent framework for understanding how finite TP Toeplitz matrices converge to infinite limits and how these limits encode representation-theoretic and geometric information.

Abstract

By a theorem of Edrei, an infinite, normalised totally nonnegative upper-triangular Toeplitz matrix is determined by a pair of nonnegative parameter sequences, the `Schoenberg parameters', where nonzero parameters correspond to the roots and poles of a naturally associated generating function. These totally nonnegative Toeplitz matrices and their parameters also arise in the classification of characters of the infinite symmetric group by later work of Thoma. Moreover the Schoenberg parameters have an asymptotic interpretation in terms of irreducible representations of S_n and their Young diagrams by Vershik-Kerov. In this article we consider infinite totally positive Toeplitz matrices as limits of finite ones, and we obtain two further asymptotic descriptions of the Schoenberg parameters that are now related to quantum cohomology of the flag variety as n goes to infinity. One is related to asymptotics of normalised quantum parameters, and the other to asymptotics of the Chern classes of the tautological line bundles. We also describe the asymptotics of (quantum) Schubert classes in terms of the Schoenberg parameters. Our limit formulas relate to and were motivated by a tropical analogue of this theory that we survey. In the tropical setting one finds an asymptotic relationship between the `tropical Schoenberg parameters' and the weight map from Lusztig's parametrisation of the canonical basis.

Totally positive Toeplitz matrices: classical and modern

TL;DR

The paper studies infinite totally positive Toeplitz matrices as limits of finite ones, anchored by Edrei's generating-function factorization, Thoma's parametrization of characters, and Vershik–Kerov's asymptotics. It develops two parallel asymptotic descriptions of Schoenberg parameters: one via quantum cohomology and Peterson's isomorphism, connecting to Schubert classes and Chern roots, and another through tropical geometry and tropical versions of Edrei's theorem, relating tropical Schoenberg parameters to Lusztig weights. The work unifies representation theory, quantum cohomology, and tropical methods to describe limiting spectral data and Schubert-theoretic behavior of Toeplitz matrices. It also provides a tropical analogue of the Edrei theory, including two versions of Edrei-type theorems and a tropical VK-type asymptotics, highlighting deep connections between algebraic combinatorics, geometry, and the canonical basis. Overall, the results give a coherent framework for understanding how finite TP Toeplitz matrices converge to infinite limits and how these limits encode representation-theoretic and geometric information.

Abstract

By a theorem of Edrei, an infinite, normalised totally nonnegative upper-triangular Toeplitz matrix is determined by a pair of nonnegative parameter sequences, the `Schoenberg parameters', where nonzero parameters correspond to the roots and poles of a naturally associated generating function. These totally nonnegative Toeplitz matrices and their parameters also arise in the classification of characters of the infinite symmetric group by later work of Thoma. Moreover the Schoenberg parameters have an asymptotic interpretation in terms of irreducible representations of S_n and their Young diagrams by Vershik-Kerov. In this article we consider infinite totally positive Toeplitz matrices as limits of finite ones, and we obtain two further asymptotic descriptions of the Schoenberg parameters that are now related to quantum cohomology of the flag variety as n goes to infinity. One is related to asymptotics of normalised quantum parameters, and the other to asymptotics of the Chern classes of the tautological line bundles. We also describe the asymptotics of (quantum) Schubert classes in terms of the Schoenberg parameters. Our limit formulas relate to and were motivated by a tropical analogue of this theory that we survey. In the tropical setting one finds an asymptotic relationship between the `tropical Schoenberg parameters' and the weight map from Lusztig's parametrisation of the canonical basis.

Paper Structure

This paper contains 19 sections, 16 theorems, 72 equations, 2 figures.

Table of Contents

  1. Introduction
  2. The Edrei Theorem
  3. The Thoma Theorem
  4. The Vershik-Kerov Theorem
  5. Finite totally positive Toeplitz matrices and asymptotics
  6. A parametrisation theorem for finite Toeplitz matrices
  7. Toeplitz matrices in standard coordinates
  8. Quantum cohomology of flag varieties
  9. Peterson theory
  10. Mirror symmetry
  11. Asymptotics of quantum parameters
  12. Asymptotics of Schubert classes
  13. The multiplication table of $\mathbb N$
  14. Tropical Toeplitz matrices
  15. The tropical analogue of $U_+$ and Lusztig's weight map
  16. ...and 4 more sections

Key Result

Theorem 1.1

A sequence $\mathbf c=(c_i)_{i=1}^\infty$ of real numbers is totally nonnegative if and only if its generating function has a factorisation of the form for some $((\boldsymbol{\alpha},\boldsymbol{\beta}),\gamma)\in\Omega_S\times\mathbb R_{\ge 0}$ where $\Omega_S$ denotes the parameter space

Figures (2)

  • Figure 1: The quiver $Q_{n+1}$
  • Figure 2: On the left, a segment of $Q_{\infty}$ and on the right the vertex labelling of the quiver $Q_{n+1}$ for $n=4$ from Definition \ref{['d:vij']}.

Theorems & Definitions (45)

  • Theorem 1.1: Edrei Theorem ASWEdrei52
  • Remark 1.2
  • Theorem 1.3: Thoma Theorem Thoma
  • Remark 1.4
  • Theorem 1.5: Vershik-Kerov Theorem VershikKerovAsymptotic
  • Theorem 2.1: rietschJAMSrietschNagoya
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • ...and 35 more