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Tropical Toeplitz matrices and parametrisations

Konstanze Rietsch

TL;DR

This work develops two parallel tropicalisations of classical parametrisations for infinite totally positive Toeplitz matrices: the Edrei–Schoenberg framework with Schoenberg parameters and the finite Lusztig–Rietsch parametrisation linked to quantum cohomology. By introducing valuative positive structures (Puiseaux series and the continuous-function semifield ${oldsymbol{ ext C}_{>0}}$) and a careful analysis of min-ideal fillings, the author builds tropical analogues for both finite and infinite cases and proves a tropical Edrei theorem while detropicalising back to Schoenberg-type data via Lusztig’s weight map. The results establish a deep bridge between Edrei–Thoma theory, Lusztig’s canonical-basis parametrisation, and tropical geometry, including asymptotic relations between finite and infinite parameters and connections to the quantum cohomology of flag varieties. The framework offers a unified view of infinite characters of $S_ z$, tropical geometry of $U_+$, and canonical-basis parametrisations, with potential implications for representation theory, combinatorics, and algebraic geometry of flag varieties.

Abstract

The set of infinite upper-triangular totally positive Toeplitz matrices has a classical parametrisation proved by Edrei et al and originally conjectured by Schoenberg, that involves pairs of sequences of positive real parameters. These matrices (and their parameters) are central for understanding characters of the infinite symmetric group by work of Thoma. On the other hand there is a very different parametrisation theorem that applies to the finite analogue of this set. These finite Toeplitz matrices and their parameters relate to quantum cohomology of flag varieties and mirror symmetry. In this paper we replace the positive reals by a semifield with valuation to then construct tropical analogues for both parametrisation theorems. In the finite case we tropicalise using positive generalised Puiseaux series. This builds on work of Judd and Lüdenbach. In the infinite case we use a new valued semifield of continuous functions. We arrive at different natural infinite analogues of totally positive Toeplitz matrices, depending on a choice of topology on our valued semifield. We then prove an asymptotic result relating the tropical parameters from the finite case to the tropicalisations of the Schoenberg parameters. Moreover, we show that our finite type tropical parametrisation map is given by Lusztig's weight map from the theory of canonical bases. This results in a surprising connection between the classical Edrei theorem with its Schoenberg parameters and Lusztig's canonical basis parametrisation.

Tropical Toeplitz matrices and parametrisations

TL;DR

This work develops two parallel tropicalisations of classical parametrisations for infinite totally positive Toeplitz matrices: the Edrei–Schoenberg framework with Schoenberg parameters and the finite Lusztig–Rietsch parametrisation linked to quantum cohomology. By introducing valuative positive structures (Puiseaux series and the continuous-function semifield ) and a careful analysis of min-ideal fillings, the author builds tropical analogues for both finite and infinite cases and proves a tropical Edrei theorem while detropicalising back to Schoenberg-type data via Lusztig’s weight map. The results establish a deep bridge between Edrei–Thoma theory, Lusztig’s canonical-basis parametrisation, and tropical geometry, including asymptotic relations between finite and infinite parameters and connections to the quantum cohomology of flag varieties. The framework offers a unified view of infinite characters of , tropical geometry of , and canonical-basis parametrisations, with potential implications for representation theory, combinatorics, and algebraic geometry of flag varieties.

Abstract

The set of infinite upper-triangular totally positive Toeplitz matrices has a classical parametrisation proved by Edrei et al and originally conjectured by Schoenberg, that involves pairs of sequences of positive real parameters. These matrices (and their parameters) are central for understanding characters of the infinite symmetric group by work of Thoma. On the other hand there is a very different parametrisation theorem that applies to the finite analogue of this set. These finite Toeplitz matrices and their parameters relate to quantum cohomology of flag varieties and mirror symmetry. In this paper we replace the positive reals by a semifield with valuation to then construct tropical analogues for both parametrisation theorems. In the finite case we tropicalise using positive generalised Puiseaux series. This builds on work of Judd and Lüdenbach. In the infinite case we use a new valued semifield of continuous functions. We arrive at different natural infinite analogues of totally positive Toeplitz matrices, depending on a choice of topology on our valued semifield. We then prove an asymptotic result relating the tropical parameters from the finite case to the tropicalisations of the Schoenberg parameters. Moreover, we show that our finite type tropical parametrisation map is given by Lusztig's weight map from the theory of canonical bases. This results in a surprising connection between the classical Edrei theorem with its Schoenberg parameters and Lusztig's canonical basis parametrisation.

Paper Structure

This paper contains 26 sections, 61 theorems, 258 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Notation and Setup
  3. Positive structure on a ring
  4. Notations for total positivity
  5. Twist map
  6. Upper-triangular Toeplitz matrices
  7. Parametrization theorem for $\operatorname{Toep}_{n+1}(\mathbf K_{>0})$
  8. Parameterization theorem for $\operatorname{UToep}_{n+1}(\mathbb R_{\min{}})$
  9. Two theorems about finite tropical Toeplitz matrices
  10. Tropical Toeplitz matrices
  11. Infinite min-ideal fillings and a tropical Edrei theorem
  12. Lusztig weight asymptotics
  13. General interpretation of the tropical Schoenberg parameters
  14. Infinite totally positive Toeplitz matrices
  15. Tropicalisations for infinite Toeplitz matrices
  16. ...and 11 more sections

Key Result

Theorem 1.1

A sequence $\mathbf c=(c_i)_{i=1}^\infty$ of real numbers is a totally nonnegative sequence if and only if its generating function has a factorisation of the form for some $(\gamma,\boldsymbol{\alpha},\boldsymbol{\beta})\in\Omega_{S}$, where $\boldsymbol{\alpha}=(\alpha_i)_{i\in\mathbb N}$ and $\boldsymbol{\beta}=(\beta_i)_{i\in\mathbb N}$.

Figures (3)

  • Figure 1: The quiver $Q_6=(\mathcal{V}_{\bullet}\cup\mathcal{V}_{\circ},\mathcal{A})$
  • Figure 2: The infinite quiver $Q_\infty$.
  • Figure 3: The quiver $Q_{n+1}$

Theorems & Definitions (189)

  • Theorem 1.1: ASWEdrei52Thoma
  • Theorem 1.2: rietschJAMSrietschNagoya
  • Theorem A
  • Theorem B
  • Theorem C
  • Conjecture A
  • Theorem D
  • Theorem E
  • Remark 1.3
  • Theorem F
  • ...and 179 more