Tropical Schoenberg Parameters

• Tropical Schoenberg parameters are defined as weakly increasing sequences (A, B) that encode infinite min-ideal fillings via Mij = min(Ai, Bj).
• They serve as the tropical analogues of classical Schoenberg parameters, providing a bridge between Toeplitz matrix factorizations and the asymptotic limits of Lusztig weights.
• Their formulation connects total positivity, tropical geometry, and canonical basis parametrization, offering a robust framework for analyzing infinite and finite Toeplitz matrices.

Tropical Schoenberg parameters are the tropical counterparts of the classical Schoenberg–Edrei parameters for totally positive Toeplitz matrices. In the tropical theory developed for infinite upper-triangular Toeplitz data, they are encoded by a pair of monotone sequences $(\mathbf A,\mathbf B)$ with entries in $\mathbb R\cup\{\infty\}$, and the associated tropical Toeplitz object is the min-ideal filling

$M_{ij}=\min(A_i,B_j).$

Their role is simultaneously structural and asymptotic: they tropicalize the classical root/pole parameters $(\boldalpha,\boldbeta)$ from Edrei’s factorization of Toeplitz generating functions, and they arise as normalized limits of finite Lusztig weights in large rank. This places them at the intersection of total positivity, tropical geometry, and canonical-basis parametrization theory (Rietsch, 29 Sep 2025, Rietsch, 8 Sep 2025).

1. Classical origin in the Edrei–Schoenberg theorem

The starting point is the classical classification of infinite upper-triangular Toeplitz matrices. Given a sequence $\mathbf c=(c_i)_{i\ge1}$, one considers the Toeplitz matrix

$$u(\mathbf c)= \begin{pmatrix} 1 & c_1 & c_2 & c_3 & \cdots\ & 1 & c_1 & c_2 & \ddots\ & & 1 & c_1 & \ddots\ & & & \ddots & \ddots \end{pmatrix},$$

equivalently $u(\mathbf c)=\bigl(c_{i-j}\bigr)_{i,j\ge 1}$ with $c_0=1$ and $c_{-n}=0$ for $n>0$. Its generating function is

$\mathbb R\cup\{\infty\}$0

Edrei’s theorem, originally conjectured by Schoenberg, states that total nonnegativity is equivalent to a factorization

$\mathbb R\cup\{\infty\}$1

for some $\mathbb R\cup\{\infty\}$2, where $\mathbb R\cup\{\infty\}$3 and $\mathbb R\cup\{\infty\}$4 are weakly decreasing, summable sequences of nonnegative real numbers. In this framework, $\mathbb R\cup\{\infty\}$5 are the Schoenberg parameters. They are the roots and poles in the factorization of the Toeplitz generating function, and they determine the matrix completely. When $\mathbb R\cup\{\infty\}$6, one obtains the Thoma simplex normalization $\mathbb R\cup\{\infty\}$7 (Rietsch, 29 Sep 2025, Rietsch, 8 Sep 2025).

The classical theory also contains an asymptotic interpretation due to Edrei. For totally positive Toeplitz matrices, certain standard coordinates $\mathbb R\cup\{\infty\}$8 built from minors converge in row and column directions to the Schoenberg parameters: $\mathbb R\cup\{\infty\}$9 This asymptotic recovery mechanism is one of the templates for the tropical theory (Rietsch, 29 Sep 2025).

2. Tropicalization and the tropical Toeplitz locus

The tropical theory replaces positive reals by a valued positive semifield. In the formulation using generalized Puiseux series or positive continuous functions, the valuation is a semifield homomorphism

$M_{ij}=\min(A_i,B_j).$0

satisfying

$M_{ij}=\min(A_i,B_j).$1

For finite Toeplitz matrices the valued field $M_{ij}=\min(A_i,B_j).$2 of generalized Puiseux series is used, with $M_{ij}=\min(A_i,B_j).$3 equal to the lowest exponent. For the infinite theory, a new positive semifield $M_{ij}=\min(A_i,B_j).$4 is introduced, where $M_{ij}=\min(A_i,B_j).$5 if $M_{ij}=\min(A_i,B_j).$6. The infinite setting requires both a strong topology and a weak topology on $M_{ij}=\min(A_i,B_j).$7 (Rietsch, 8 Sep 2025).

The finite positive geometry comes from the unipotent group $M_{ij}=\min(A_i,B_j).$8. In Lusztig’s standard positive chart,

$M_{ij}=\min(A_i,B_j).$9

and tropicalization sends the standard coordinates to

$(\boldalpha,\boldbeta)$0

Within these tropical coordinates, the Toeplitz locus is characterized by the min-ideal condition

$(\boldalpha,\boldbeta)$1

for all interior positions. Equivalently, in root language,

$(\boldalpha,\boldbeta)$2

This is the tropical analogue of the Toeplitz condition. The resulting tropical Toeplitz locus is canonical in the sense that it is independent of the choice of reduced word for $(\boldalpha,\boldbeta)$3 (Rietsch, 29 Sep 2025).

3. Definition and parametrization of tropical Schoenberg parameters

The tropical Schoenberg parameters are the infinite tropical replacement for the classical parameter sequences $(\boldalpha,\boldbeta)$4. In the 2025 tropical Toeplitz literature, they are given by two weakly increasing sequences

$(\boldalpha,\boldbeta)$5

with values in $(\boldalpha,\boldbeta)$6 (Rietsch, 29 Sep 2025, Rietsch, 8 Sep 2025).

Two closely related parameter-space formulations appear.

Aspect	Classical Schoenberg data	Tropical Schoenberg data
Parameters	weakly decreasing $(\boldalpha,\boldbeta)$7	weakly increasing $(\boldalpha,\boldbeta)$8
Ambient values	nonnegative real numbers	$(\boldalpha,\boldbeta)$9
Reconstruction	factorization of generating function	$\mathbf c=(c_i)_{i\ge1}$0

One formulation imposes a weak interlacing condition

$\mathbf c=(c_i)_{i\ge1}$1

Another packages the infinite tropical parameter space as

$\mathbf c=(c_i)_{i\ge1}$2

together with subspaces $\mathbf c=(c_i)_{i\ge1}$3 for interlacing sequences and $\mathbf c=(c_i)_{i\ge1}$4 when all entries are real (Rietsch, 29 Sep 2025, Rietsch, 8 Sep 2025).

From such a pair one forms the filling

$\mathbf c=(c_i)_{i\ge1}$5

This defines the tropical Edrei map

$\mathbf c=(c_i)_{i\ge1}$6

The central structural statement is that this map is bijective: $\mathbf c=(c_i)_{i\ge1}$7 where $\mathbf c=(c_i)_{i\ge1}$8 denotes the space of infinite min-ideal fillings. There is also a stable version

$\mathbf c=(c_i)_{i\ge1}$9

and in this stable setting interlacing corresponds exactly to stabilization of rows and columns (Rietsch, 29 Sep 2025, Rietsch, 8 Sep 2025).

4. Valuation-theoretic interpretation

A defining feature of tropical Schoenberg parameters is that they are not merely formal tropical coordinates; they are the valuations of classical Schoenberg parameters. In a valued setting, one lifts a tropical pair $$u(\mathbf c)= \begin{pmatrix} 1 & c_1 & c_2 & c_3 & \cdots\ & 1 & c_1 & c_2 & \ddots\ & & 1 & c_1 & \ddots\ & & & \ddots & \ddots \end{pmatrix},$$0 to actual positive parameters, for example by

$$u(\mathbf c)= \begin{pmatrix} 1 & c_1 & c_2 & c_3 & \cdots\ & 1 & c_1 & c_2 & \ddots\ & & 1 & c_1 & \ddots\ & & & \ddots & \ddots \end{pmatrix},$$1

so that

$$u(\mathbf c)= \begin{pmatrix} 1 & c_1 & c_2 & c_3 & \cdots\ & 1 & c_1 & c_2 & \ddots\ & & 1 & c_1 & \ddots\ & & & \ddots & \ddots \end{pmatrix},$$2

The associated classical generating series is

$$u(\mathbf c)= \begin{pmatrix} 1 & c_1 & c_2 & c_3 & \cdots\ & 1 & c_1 & c_2 & \ddots\ & & 1 & c_1 & \ddots\ & & & \ddots & \ddots \end{pmatrix},$$3

or, in the infinite-valued setting,

$$u(\mathbf c)= \begin{pmatrix} 1 & c_1 & c_2 & c_3 & \cdots\ & 1 & c_1 & c_2 & \ddots\ & & 1 & c_1 & \ddots\ & & & \ddots & \ddots \end{pmatrix},$$4

The Toeplitz matrix built from this series has standard coordinates whose valuations recover the tropical filling: $$u(\mathbf c)= \begin{pmatrix} 1 & c_1 & c_2 & c_3 & \cdots\ & 1 & c_1 & c_2 & \ddots\ & & 1 & c_1 & \ddots\ & & & \ddots & \ddots \end{pmatrix},$$5 Thus the tropical Toeplitz matrix is literally the valuation image of the classical one (Rietsch, 29 Sep 2025).

A more refined description uses supersymmetric Schur functions. For $$u(\mathbf c)= \begin{pmatrix} 1 & c_1 & c_2 & c_3 & \cdots\ & 1 & c_1 & c_2 & \ddots\ & & 1 & c_1 & \ddots\ & & & \ddots & \ddots \end{pmatrix},$$6, the standard coordinates satisfy

$$u(\mathbf c)= \begin{pmatrix} 1 & c_1 & c_2 & c_3 & \cdots\ & 1 & c_1 & c_2 & \ddots\ & & 1 & c_1 & \ddots\ & & & \ddots & \ddots \end{pmatrix},$$7

and the valuation formula

$$u(\mathbf c)= \begin{pmatrix} 1 & c_1 & c_2 & c_3 & \cdots\ & 1 & c_1 & c_2 & \ddots\ & & 1 & c_1 & \ddots\ & & & \ddots & \ddots \end{pmatrix},$$8

implies

$$u(\mathbf c)= \begin{pmatrix} 1 & c_1 & c_2 & c_3 & \cdots\ & 1 & c_1 & c_2 & \ddots\ & & 1 & c_1 & \ddots\ & & & \ddots & \ddots \end{pmatrix},$$9

This provides a direct valuation-theoretic derivation of the tropical Edrei map (Rietsch, 8 Sep 2025).

5. Asymptotic recovery from Lusztig’s weight map

The asymptotic significance of tropical Schoenberg parameters lies in their recovery from finite tropical Toeplitz data. Let

$u(\mathbf c)=\bigl(c_{i-j}\bigr)_{i,j\ge 1}$0

be a sequence of finite min-ideal fillings converging uniformly to an asymptotically real infinite filling

$u(\mathbf c)=\bigl(c_{i-j}\bigr)_{i,j\ge 1}$1

Write the Lusztig weight of the finite filling as

$u(\mathbf c)=\bigl(c_{i-j}\bigr)_{i,j\ge 1}$2

Then the normalized edge coordinates of the diagonal converge to the tropical Schoenberg parameters: $u(\mathbf c)=\bigl(c_{i-j}\bigr)_{i,j\ge 1}$3 The left end of the diagonal converges to the $u(\mathbf c)=\bigl(c_{i-j}\bigr)_{i,j\ge 1}$4, while the right end, with the longest-element involution $u(\mathbf c)=\bigl(c_{i-j}\bigr)_{i,j\ge 1}$5 and the corresponding minus sign, converges to the $u(\mathbf c)=\bigl(c_{i-j}\bigr)_{i,j\ge 1}$6. This is the tropical analogue of a Vershik–Kerov type asymptotic statement (Rietsch, 29 Sep 2025, Rietsch, 8 Sep 2025).

The same result can be phrased by saying that the normalized finite map

$u(\mathbf c)=\bigl(c_{i-j}\bigr)_{i,j\ge 1}$7

converges, in the limit, to the inverse of the tropical Edrei map. In particular, the tropical Schoenberg parameter pair $u(\mathbf c)=\bigl(c_{i-j}\bigr)_{i,j\ge 1}$8 is asymptotically extracted from finite Lusztig weight data. The finite tropical parametrization itself is Lusztig’s weight map, and the relevant tropicalization of the finite Toeplitz parametrization map is identified with this canonical-basis weight map (Rietsch, 29 Sep 2025, Rietsch, 8 Sep 2025).

The asymptotic theorem requires the asymptotically real hypothesis. A plausible implication is that tropical Schoenberg parameters are robust only within a controlled asymptotic regime; without that hypothesis, the normalized finite weights need not converge to the expected infinite tropical data (Rietsch, 8 Sep 2025).

6. Mathematical scope and distinction from other Schoenberg notions

Tropical Schoenberg parameters mediate a three-way connection emphasized in the 2025 literature: classical infinite totally nonnegative Toeplitz matrices and their Edrei–Schoenberg factorization, finite totally positive Toeplitz matrices and Lusztig/Peterson parametrizations from flag varieties, and a tropical limit in which Toeplitz data become min-ideal fillings and the parameterization becomes piecewise-linear (Rietsch, 29 Sep 2025). In the companion treatment of tropical Toeplitz matrices, this bridge is also tied to quantum cohomology of the flag variety, mirror symmetry, and the historical role of Schoenberg parameters in the theory of characters of the infinite symmetric group (Rietsch, 8 Sep 2025).

A common source of confusion is the coexistence of several distinct “Schoenberg” theories on arXiv. Tropical Schoenberg parameters are not the Schoenberg coefficients appearing in isotropic positive definite kernels on spheres. In the spherical setting, Schoenberg’s theorem and its extensions concern nonnegative spectral expansions in Gegenbauer or tensor Gegenbauer bases; for example, on $u(\mathbf c)=\bigl(c_{i-j}\bigr)_{i,j\ge 1}$9 the isotropic part has a double Gegenbauer expansion with nonnegative coefficients and a double summability condition (Guella et al., 2015). Related works study $c_0=1$0-Schoenberg coefficients, dimension-walk relations, and strict positive definiteness on real and complex spheres (Arafat et al., 2018, Bissiri et al., 2018). Those are spectral coefficients for positive definite kernels, whereas tropical Schoenberg parameters are valuation-level invariants for Toeplitz total positivity.

The resulting picture is specific and rigid. Tropical Schoenberg parameters are monotone sequence data $c_0=1$1, their associated tropical Toeplitz matrices are exactly the min-ideal fillings $c_0=1$2, and their principal significance is that they simultaneously tropicalize the classical Edrei–Schoenberg factorization and encode the large-rank limit of Lusztig weight data (Rietsch, 29 Sep 2025, Rietsch, 8 Sep 2025).