Meetings: On Zoom MWF 2:30 -- 3:20.
Content: We will cover the (related) subjects of total positivity and cluster algebras. Classically, total positivity concerns matrices whose minors are all nonnegative. We will discuss this and also more modern incarnations and applications. Cluster algebras are a class of commutative algebras with an elaborate recursive definition. They were introduced about 20 years ago, and have connections with diverse areas of mathematics (cf. the surveys below). We will discuss the key structural theorems and key examples. At the end of the semester, we will discuss an advanced topic (scattering diagrams).
Prereqs: Self-contained; familiarity with root systems & Lie/rep theory would be helpful context.
Homework: Three problem sets, mandatory for undergraduates, recommended for early graduate students. 1st HW: solve 5 problems from this list by Oct. 7. 2nd HW: solve 4 more problems by Nov. 9. Final HW: solve 3 more problems from the list by Wed., Dec 16.
Lecture notes: Trevor Karn compiled a list of the main results from the total positivity portion of the class.
9/9. TNN cell decomposition of U_3. Notes.
9/11. TP for 2 by 2 minors via triangulations. Notes.
9/14. TP matrices and eigenvalues, TNN matrices and roots of polynomials / analytic functions. Notes.
9/16. TP and sign variation; Cauchy-Binet; TNN matrices admit TNN LDU factorizations. Notes.
9/18. TNN unipotent semigroup is is generated by Chevalley generators; reduced words for permutations. Notes.
9/21. "Chevalley factorizations" from reduced words; Bruhat decomposition for GLn. Notes.
9/23. TNN cells are disjoint and Chevalley factorization is a homeo. Cell closure order is Bruhat order on permutations. Notes.
9/25. TNN cells in other Lie types; Lindstrom lemma + network description of TNN matrices. Notes.
9/28. Initial (solid) minors are a TP test. Notes.
9/30. Row-solid minors are a TNN test; unipotent TP tests from wiring diagrams Notes.
10/2. GLn TP tests from double wiring diagrams; inverting the Chevalley factorization map. Notes.
10/5. Quivers from wiring diagrams. Notes.
10/7. Quiver mutation. Notes.
10/9. Seed mutation. Notes.
10/12. Definition of cluster algebra. Notes.
10/14. Laurent phenomenon, denominator vectors, Somos sequences. Notes.
10/16. Finite cluster type(s). Notes.
10/19. Grassmannians. Notes.
10/21. Gr(2,n) and type A cluster algebras. Frieze patterns. Notes.
10/23. Cluster monomials; noncrossing vs standard monomials for Gr(2,n). Notes.
10/26. "Canonical" bases for G-reps; decomposing coordinate rings into G-irreps. Notes.
10/28. Finite type classsification: rank 2. Notes.
10/30. Laurent phenomenon in rank 2; lower + upper bounds; upper cluster algebra. Notes.
11/2. The Starfish Lemma. Notes.
11/4. Applying the starfish lemma. Notes.
11/6. Cluster structure on Gr(k,n). Notes.
11/9. Triangulations of bordered marked surfaces. Notes.
11/11. Cluster algebras from surfaces. Notes.
11/13. Finite mutation types; mapping class groups and cluster algebras. Notes.
11/16. Bases for cluster algebras from surfaces. Notes.
11/18. Positroids. Notes.
11/20. Positroids, Grassmann necklaces, decorated permutations, and chord diagrams. Cell closure order. Notes.
11/23. Reduction lemma for positroid cells. Notes.
11/25. Plabic graphs, boundary measurement map. Notes.
11/30. Surjectivity of the boundary measurement map; trip permutations; move-connectedness for plabic graphs. Notes.
12/2. Seeds from plabic graphs. Notes.
12/4. Tensor diagrams as cluster variables in Grassmannians. Notes.
12/7. Arborizability; SLk skein relations; d-vector cones. Notes.
12/9. g-vectors and g-vector fans. Notes.
12/11. Sign-coherence; tropical duality between g- and c- vectors; walls. Notes.
12/14. Wall-crossing automorphisms and consistent scattering diagrams. Notes.
12/16. Broken lines and theta functions. Notes.
References:
Textbook: Introduction to Cluster Algebras chapters 1--3, 4-5, and 6 (Fomin, Williams, and Zelevinsky).
TP Surveys:
Total positivity: tests and parametrizations (Fomin & Zelevinsky)
Lecture notes on TP (Pylyavskyy)
A survey of total positivity (Lusztig)
Cluster surveys:
Cluster algebras: an introduction (Williams)
More specialized cluster surveys:
Cluster algebras and representation theory (Leclerc), Littlewood-Richardson to CA (Zelevinsky), or Crystal bases and categorification (Kashiwara)
Cluster categories (Reiten) or any of these nice surveys (Keller)
this search leads to other such surveys.
Links:
Cluster algebras portal , Bernhard Keller's Quiver mutation applet , and Paval Galashin's plabic tilings applet