Total positivity

Summer Semester 2017


A matrix is called totally positive (resp., totally non-negative) if all its minors are positive (resp., non-negative). These classes of matrices arise and play an important role in various domains of mathematics, such as: representation theory, cluster algebras, combinatorics, probability and stochastic processes, mathematical physics, to name but a few. The goal of this lecture is to provide an introduction to total positivity, emphasized on the algebraic, combinatorial and geometric aspects on this subject.


Grade from final exam


  • (31.07.2017) The grade are now available. The Klausureinsicht will be on Wednesday 2, August 14h in Room 2.10.
  • (13.07.2017) The Exercise 3 in Sheet 6 is updated. The old version contains mistakes in the description of the network (sorry!).
  • (07.07.2017) The registration of the exam is now possible. Please find the information I received from Mrs. Georg below: Die Studierenden sollen bitte entsprechende Formulare ausfüllen (zu finden für die jeweiligen Studiengänge und Versionen der Prüfungsordnungen unter und diese bei mir im Geschäftszimmer abgeben. Anträge auf schriftliche Prüfung können ebenfalls bei mir eingereicht werden. Die Anmeldefrist läuft bis eine Woche vor dem Prüfungstermin, d.h. bis zum 20.07.2017. An diesem Tag endet auch die Rücktrittsfrist.
  • (25.04.2017) The exercise class will be on Thursday 17h45-19h15, Seminarraum 2 (first class: 04.05.2017).
  • (18.04.2017) The exam will be on 27.07.2017, 13h45-15h45.
  • (18.04.2017) The first exercise session will be held in the week of 01.05.2017, time and place to be announced.
  • (18.04.2017) The lectures will be on Thursday 14h-15h30 and Friday 10h-11h30 in Seminarraum 1. Office hour will be on Thursday 15h45-16h30 or by appointment.


    Exercise Sheet 1 (24.04.2017) will be discussed on 04.05.2017, please hand in your exercises in the lecture of 04.05.2017. (In Exercise 1, the definition of A is clarified.)
    Exercise Sheet 2 (05.05.2017) will be discussed on 18.05.2017, please hand in your exercises in the lecture of 18.05.2017.
    Exercise Sheet 3 (19.05.2017) will be discussed on 01.06.2017, please hand in your exercises in the lecture of 01.06.2017.
    Exercise Sheet 4 (02.06.2017) will be discussed on 22.06.2017, please hand in your exercises in the lecture of 22.06.2017.
    Exercise Sheet 5 (23.06.2017) will be discussed on 06.07.2017, please hand in your exercises in the lecture of 06.07.2017.
    Exercise Sheet 6 (07.07.2017) will be discussed on 20.07.2017, please hand in your exercises in the lecture of 20.07.2017.


  • Lecture 1 (20.04.2017): Introduction, wiring diagrams, determinants.
  • Lecture 2 (21.04.2017): Immanants, Laplace expansion, Schur complements. (The theorem of Stembridge on immanants is proved in this paper: J. Stembridge, Immanants of totally positive matrices are nonnegative, Bull. London Math. Soc. 23 (1991), 422--428. PDF)
  • Lecture 3 (27.04.2017): Sylvester identity, Jacobi formula, Cauchy-Binet.
  • Lecture 4 (28.04.2017): Pluecker relations, totally positive matrices.
  • Lecture 5 (04.05.2017): Examples of TP matrices.
  • Lecture 6 (05.05.2017): More examples, Fekete Lemma.
  • Lecture 7 (11.05.2017): Fekete theorem and applications, Gasca-Pena criterion.
  • Lecture 8 (12.05.2017): Proof of Gasca-Pena, density and TNN criterion.
  • Lecture 9 (18.05.2017): LDU decomposition.
  • Lecture 10 (19.05.2017): Triangular TP and TNN, factorization.
  • Lecture 11 (26.05.2017): Loewner-Whitney theorem.
  • Lecture 12 (01.06.2017): Free groups.
  • Lecture 13 (02.06.2017): Symmetric groups: generators and relations.
  • Lecture 14 (16.06.2017): Reduced decompositions.
  • Lecture 15 (22.06.2017): Tits' theorem, longest element in symmetric group.
  • Lecture 16 (23.06.2017): Symmetric functions.
  • Lecture 17 (29.06.2017): Young tableaux, graphs, networks.
  • Lecture 18 (30.06.2017): Lindstörm Lemma.
  • Lecture 19 (06.07.2017): Applications of Lindström Lemma (total positivity).
  • Lecture 20 (07.07.2017): Applications of Lindström Lemma (total positivity).
  • Lecture 21 (13.07.2017): Applications of Lindström Lemma (total positivity), revision.
  • Lecture 22 (14.07.2017): Applications of Lindström Lemma (symmetric functions).
  • Lecture 23 (20.07.2017): Double wiring diagrams and factorisation schemes.
  • Lecture 24 (21.07.2017): Geometric crystals.


  • T. Ando, Totally positive matrices, Linear Algebra Appl. 90 (1987), 165-219.
  • N. Arkani-Hamed, J. Bourjaily, F. Cachazo, A. Goncharov, A. Postnikov, J. Trnka, Scattering Amplitudes and the Positive Grassmannian,
  • A. Berenstein, S. Fomin and A. Zelevinsky, Parametrizations of canonical bases and totally positive matrices, Advances in Mathematics 122 (1996), 49-149.
  • A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math., vol. 143, 1 (2001), 77-128.
  • S. Fomin and A. Zelevinsky, Total positivity: tests and parametrizations, Math. Intelligencer 22 (2000), 23-33.
  • S. Lang, Algebra (3rd ed), Graduate Texts in Mathematics 211, Springer Verlag.
  • P. Littelmann, Bases canoniques et applications, Séminaire Bourbaki, 1997-1998, Exp.No. 847, pp. 287-306.
  • G. Lusztig, Introduction to total positivity, de Gruyter Exp. Math. 26 (1998), 133-145.
  • A. Pinkus, Totally Positive Matrices, Cambridge University Press, 2009, ISBN 9780521194082.
  • A. Postnikov, Total positivity, Grassmannians, and networks, available at
  • P. Pylyavskyy, Lecture notes on total positivity.

  • Arbeitsgruppe Algebra und Zahlentheorie, Mathematisches Institut, Universität zu Köln