Math 665 Fall 2013
Total positivity

Lectures: MWF 11-12

Instructor: Thomas Lam, 2834 East Hall, tfylam@umich.edu

Office Hours: by appointment.

Prerequisites:
I will assume familiarity with linear algebra and this is the only prerequisite in the beginning. Towards the later half of the course, it will be helpful to be familiar with (1) Coxeter groups and root systems, and (2) semisimple complex algebraic groups.

Grading: There will be weekly-biweekly problem sets. There will be a term paper roughly 5 typed pages in length which can either (1) be expository, or (2) explore one of the open problems discussed in class. There will be no final exam.

Lecture notes: Notes for TNN part of Grassmannian (very rough)

Announcement:

Problem sets: Problem sets are posted here and are due in class on the stated date. Starred problems can be handed in anytime before December 6.
Some of the starred problems, particularly the open ones are suitable for the term paper. You are welcome to work in groups of 2-3 (but no bigger!) if you choose to do this.
Problem Set 1 (Due Friday September 13) Last modified: September 4 Comments
Problem Set 2 (Due Wednesday September 25) Last modified: September 13
Problem Set 3 (Due Friday October 4) Last modified: September 30 (there was a typo in Problem 2, part 2; thanks Chris for pointing this out!)
Problem Set 4 (Due Friday October 11) Last modified: October 4
Problem Set 5 (Due Friday October 25) Last modified: October 11 Solution to Problem 4 part (4)
Problem Set 6 (Due Wednesday November 6) Last modified: October 28 (A few typos were fixed thanks to Gabe)
Problem Set 7 (Due Friday November 15) Last modified: November 6

Lecture summary:

• Lecture 1 (09/04): Three equivalent descriptions of TNN part of GL_n
• Lecture 2 (09/06): Lindstrom lemma, TP part via special networks and initial minors
• Lecture 3 (09/09): Reduction of TNN part of GL_n to semigroup generators
• Lecture 4 (09/11): Cell decomposition of TNN part of U. Summary of some properties of S_n (generators, relations, Bruhat order)
• Lecture 5 (09/13): Polynomial equations for Bruhat cells. The claim I made about reduced words connected by braid moves is less clear than I hoped. For a better proof see Theorem 1.1.2 here.
• Lecture 6 (09/16): Proved cell decomposition of TNN part of U, and closure order is given by Bruhat order.
• Lecture 7 (09/18): Totally positive functions. Relation to characters of infinite symmetric group. A good reference for character theory of infinite symmetric group is [Ker]
• Lecture 8 (09/23): Quick run through symmetric functions and Frobenius characteristic map. Relation between harmonic functions on Young's graph and characters of infinite symmetric group.
• Lecture 9 (09/25): Proved extremal harmonic functions on Young's graph corresponds to totally positive homomorphisms of symmetric functions. A discussion of two special totally positive homomorphisms.
• Lecture 10 (09/27): Started proof of Edrei-Thoma theorem.
• Lecture 11 (09/30): Finished proof of Edrei-Thoma theorem. Thoma multiplicativity for characters.
• Lecture 12 (10/02): Started discussing total positivity for loop groups. Reduction of GL_n case to unipotent case.
• Lecture 13 (10/04): Discuss TNN part of polynomial loop group. Started discussing cylindrical networks.
• Lecture 14 (10/07): Cylindric Lindstrom Lemma.
• Lecture 15 (10/09): Whirl and curl matrices. First factorization theorem for TNN part of formal loop group.
• Lecture 16 (10/11): Second factorization theorem involving Chevalley generators. R-matrix for whirls and curls.
• Lecture 17 (10/16): R-matrix via networks.
• Lecture 18 (10/18): R-matrices and uniqueness in factorization theorem of Lecture 15.
• Lecture 19 (10/21): Guest lecture by Suho Oh on positroids.
• Lecture 20 (10/23): Infinite reduced words, and corresponding parametrizations x_i.
• Lecture 21 (10/25): Reduced TNN braid limit theorem to TNN exchange lemma.
• Lecture 22 (10/28): Discussion of TNN exchange lemma. Introduced weak order, and affine permutation model for affine symmetric group.
• Lecture 23 (10/30): Matchings of bipartite graphs in a disk and their boundary measurements
• Lecture 24 (11/01): Boundary measurements satisfy the Plucker relation
• Lecture 25 (11/04): Relations on bipartite graphs preserving measurements
• Lecture 26 (11/06): Trips and zigzag paths
• Lecture 27 (11/08): Bounded affine permutations
• Lecture 28 (11/11): Matroids and Grassmann necklaces
• Lecture 29 (11/13): Bridge decompositions 1
• Lecture 30 (11/15): Bridge decompositions 2
• Lecture 31 (11/18): Every TNN point in Grassmannian is representable. Classification of positroids.
• Lecture 32 (11/20): Electrical networks and response matrices
• Lecture 33 (11/22): Matrix-Tree theorem, and grove measurements
• Lecture 34 (11/25): Local equivalences, medial graphs
• Lecture 35 (11/27): Statement of main theorems concerning circular planar electrical networks
• Lecture 36 (12/2): The construction of a planar bipartite graph from an electrical network
• Lecture 37 (12/4): The map from groves to almost perfect matchings
• Lecture 38 (12/6): Cactus networks and their medial graphs
• Lecture 39 (12/9): Partial order on medial pairings, and the affine Bruhat order
• Lecture 40 (12/11): More about the partial order on medial pairings

References:
[Pin] A. Pinkus, Totally Positive Matrices, Cambridge Tracts in Mathematics
[Lus] G. Lusztig, Total positivity in reductive groups, in "Lie theory and geometry", Progress in Mathematics, 123, Birkhauser Boston 1994, 531-568.
[Kar] S. Karlin, Total positivity, Volume 1, Stanford University Press.
[FZ] S. Fomin and A. Zelevinsky, Total positivity: tests and parametrizations, The Mathematical Intelligencer, 22 (2000), 23-33.
[Bre] F. Brenti, Unimodal, Log-concave and Polya Frequency Sequences in Combinatorics, Memoirs AMS 413 (1989).
[Pos] A. Postnikov, Total positivity, Grassmannians, and networks
[LP] T. Lam and P. Pylyavskyy Total positivity in loop groups I: whirls and curls, Adv. in Math. 230 (2012), 1222-1271.
[BZ] A. Berenstein and A. Zelevinsky, Tensor product multiplicities, canonical bases and totally positive varieties, Invent. Math. 143 (2001), 77-128.
[Pyl] P. Pylyavskyy, Course on total positivity
[GK] A. Goncharov and R. Kenyon Dimers and cluster integrable systems
[Ker] S. Kerov, Asymptotic representation theory of the symmetric group and its applications in analysis

List of topics (tentative and potentially ambitious):
Core topics:

• Total nonnegativity for GL_n
• Fekete's criterion for total positivity
• Semigroup description of the totally nonnegative part of GL_n
• Planar acyclic networks and Lindstrom Lemma
• Upper triangular total positivity and parametrizations
• Totally positive functions and Edrei-Thoma theorem
• Plabic graphs and total nonnegativity for Grassmannians
• Total positivity for loop groups and networks on surfaces

• Stembridge's work on irreducible immanants
• Thoma's work on irreducible characters of infinite symmetric group
• Lusztig and Fomin-Berenstein-Zelevinsky's work on parametrizations of canonical bases
• Fomin and Zelevinsky's work on totally positive bases and cluster algebras