Total positivity

2026-10-22 Improve this page

#Total positivity and total nonnegativity

Total positivity is the study of matrices and kernels whose minors have a fixed sign. A matrix is totally nonnegative (TNN) if all minors are nonnegative, and totally positive if all minors are positive. See [Kar68, FJ11] for general background.

This page is a navigation point for the versions that occur most often in algebraic combinatorics: planar networks, Pólya frequency sequences, real zeros, and totally nonnegative Grassmannians.

Example

The matrix \[\begin{pmatrix} 1 & 1\\ 1 & 2 \end{pmatrix}\] is totally positive: all entries are positive, and its determinant is $1.$ For larger matrices, checking all minors is usually inefficient, so one uses structure such as planar networks, Toeplitz form, or factorization.

#Planar networks

The Lindström–Gessel–Viennot lemma explains why planar networks produce TNN matrices. If a matrix entry counts weighted paths from a source to a sink in a compatible planar network, then each minor counts families of nonintersecting paths and is therefore nonnegative.

If $M_{i,j}$ is the weighted path sum from source $s_i$ to sink $t_j,$ then a $2\times2$ minor has the form \[M_{i,a}M_{j,b}-M_{i,b}M_{j,a}.\] The LGV involution cancels the intersecting path pairs, leaving only nonintersecting pairs. In a planar compatible network, the remaining weights are nonnegative. This is the local mechanism behind total nonnegativity of the whole path matrix.

This mechanism appears in:

path matrices for total positivity;
Jacobi–Trudi and flagged Jacobi–Trudi determinants;
planar networks and plabic graphs for positroids;
network proofs of real-rootedness and interlacing.

#Pólya frequency sequences

A sequence $a=(a_0,a_1,\dotsc)$ is a Pólya frequency sequence if its Toeplitz matrix $T(a)_{i,j}=a_{i-j}$ is TNN. The Aissen–Schoenberg–Whitney theorem identifies finite Pólya frequency sequences with polynomials whose zeros are real and nonpositive.

Thus total positivity gives one of the main bridges between coefficient sequences and real-rooted polynomials. It also explains closure properties: products of TNN matrices are TNN, so convolutions of Pólya frequency sequences remain Pólya frequency sequences.

#Grassmannians and positroids

For a $k\times n$ matrix $A,$ the maximal minors $\Delta_I(A)$ are the Plücker coordinates of a point of the Grassmannian $\mathrm{Gr}(k,n).$ The totally nonnegative Grassmannian consists of points with all Plücker coordinates nonnegative. Its cells are indexed by positroids, introduced by A. Postnikov [Pos06].

This is a different but closely related face of total positivity: instead of asking all minors of a square matrix to be nonnegative, one fixes the maximal minors that define a Grassmannian point.

#Real-rootedness

Total positivity frequently packages root-location statements. The Pólya frequency page treats Toeplitz total positivity and real-rootedness of coefficient sequences. The interlacing page uses total nonnegativity to encode pairs and families of interlacing polynomials. The stable-polynomial page gives a multivariate framework that often implies the same univariate real-rootedness results after specialization.

In practice, the useful question is often not "are all minors nonnegative?" but "which structured matrix has the desired minors?" Once the matrix is identified as a path matrix, Toeplitz matrix, or product of elementary TNN matrices, the desired positivity follows from closure properties.

#Hankel matrices and continued fractions

A polynomial sequence $(a_n)_{n\geq0}$ is coefficientwise Hankel-totally nonnegative if every minor of the Hankel matrix \[\bigl(a_{i+j}\bigr)_{i,j\geq0}\] has nonnegative coefficients. Stieltjes and Jacobi continued fractions often prove this property by realizing $(a_n)$ as the moments of weighted paths.

C. Xu and J. Zeng derive Jacobi continued fractions for multivariate $(p,q)$-refinements of bi-Stirling Eulerian polynomials [XZ26]. Their weighted Motzkin-path models give gamma-positive expansions and prove coefficientwise Hankel total nonnegativity for two alternating-permutation families and for a generalized Eulerian family [XZ26, Thms. 7.2, 7.4, and 7.7].

For fixed $0\lt{}q\lt{}1$ and $-1\lt{}\beta\lt{}1,$ J. Zeng proves coefficientwise Hankel total nonnegativity for affine Rogers moments and their associated Jacobi tails. His second $q$-deformation gives a Narayana kernel whose triangle is TNN for $\theta\gt{}-1,$ while its associated square is strictly totally positive [Zen26, Thm. 2.1 and Thms. 12.1, 12.3].

Bibliography

[FJ11]Shaun M. Fallat and Charles R. Johnson. Totally nonnegative matrices. Princeton series in applied mathematics. Princeton University Press, 2011.

.bib

@book{FallatJohnson2011,
  author    = {Fallat, Shaun M. and Johnson, Charles R.},
  title     = {Totally Nonnegative Matrices},
  publisher = {Princeton University Press},
  series    = {Princeton Series in Applied Mathematics},
  year      = {2011},
  doi       = {10.1515/9781400839018}
}

[Kar68]Samuel Karlin. Total positivity, Vol. I. Stanford University Press, 1968.

.bib

@book{Karlin1968,
  author    = {Karlin, Samuel},
  title     = {Total Positivity, {V}ol.~{I}},
  publisher = {Stanford University Press},
  year      = {1968}
}

[Pos06]Alexander Postnikov. Total positivity, Grassmannians, and networks. arXiv:math/0609764, 2006.

.bib

@article{Postnikov2006x,
Author = {Alexander Postnikov},
Title = {Total positivity, {G}rassmannians, and networks},
Year = {2006},
Eprint = {math/0609764},
  url = {https://arxiv.org/abs/math/0609764},
journal = {arXiv e-prints}
}

[XZ26]Chao Xu and Jiang Zeng. The $(p,q)$-analogues of bi-Stirling Eulerian polynomials via continued fractions. arXiv:2609.14386v1, 2026.

.bib

@article{XuZeng2026x,
  author = {Chao Xu and Jiang Zeng},
  title = {The $(p,q)$-analogues of bi-{S}tirling {E}ulerian polynomials via
    continued fractions},
  year = {2026},
  eprint = {2609.14386v1},
  url = {https://arxiv.org/abs/2609.14386v1},
  journal = {arXiv e-prints}
}

[Zen26]Jiang Zeng. Two q-Deformations of Narayana Polynomials: Rogers Moments, Hankel Positivity, and q-Narayana Kernels. arXiv:2610.28691v1, 2026.

.bib

@article{Zeng2026qNarayana,
  author = {Jiang Zeng},
  title = {Two q-{D}eformations of {N}arayana {P}olynomials: {R}ogers {M}oments, {H}ankel {P}ositivity, and q-{N}arayana {K}ernels},
  year = {2026},
  eprint = {2610.28691v1},
  url = {https://arxiv.org/abs/2610.28691v1},
  journal = {arXiv e-prints}
}