The symmetric group (Projects)

bijective proof
Viennot's construction
Permutohedron
Task
Learn a new theorem in combinatorics, group theory, or representation theory and communicate it in written form.
Minimum Requirements
  • Each student will focus on a different result.
  • The written document will introduce/motivate, correctly state, and prove a theorem. It will also include at least one interesting example, construction, or special case illustrating the theorem. The article will be as self-contained as possible. The new document must be typed, be at most eight pages in length (with one inch margins and a 12pt font), and be available in PDF format.
Assessment
Project grades will be computed as follows:
Due Date Element Weight
2020–01–24 research 10%
2020–02–07 outline 10%
2020–03–06 rough draft 10%
2020–03–20 feedback 20%
2020–04–03 project 50%
Advice
Paul R. Halmos and Steven L. Kleiman each provide some suggestions on how to present mathematics.
Comments
By design, this assignment is very open-ended. Students are strongly encouraged to explore examples. Students are also encouraged to formulate, test, and prove their own conjectures. Here are some questions that you may want to consider:
  • What was the original motivation or historical context for your theorem?
  • Can you give more than one proof of your theorem?
  • Does your theorem have any interesting specializations or important applications?
  • Can your theorem be generalized?
Potential Topics
The following are natural candidates:
  1. Alternating groups — SAMUEL GRAY
  2. Alternating sign matrices
  3. Artin-Wedderburn theorem
  4. Association schemes
  5. Braid groups — HANNAH SNETSINGER
  6. Bruhat order
  7. Cartan-Dieudonné theorem — KIERA PARTRIDGE
  8. Exceptional outer automorphisms
  9. Field with one element (permutations are flags)
  10. Gale-Ryser theorem
  11. Gelfand pair
  12. Gelfand-Tsetlin pattern
  13. Group actions on boolean algebras
  14. Hyperoctahedral groups
  15. Inversions — GABBY WOLFE
  16. Jordan's theorem — JACK MCKINNON
  17. Pattern-avoiding permutations
  18. Pólya theory — ELIZA NAVERA
  19. Permutohedron — DANIEL BARAKE
  20. Random permutations
  21. Reflection groups
  22. Representation theory of the special unitary group
  23. Schreier-Sims algorithm
  24. Schur polynomials
  25. Schur-Weyl duality
  26. Sorting algorithms
  27. Steinhaus-Johnson-Trotter algorithm
  28. Shuffling cards — EDII MASON
  29. Symmetric polynomials — ADAM PAQUETTE
  30. Todd-Coxeter algorithm
  31. Unimodality
  32. Wallpaper groups — CURTIS WILSON
  33. Wreath products (of symmetric groups) — BENJAMIN SYMS-WILSON
  34. Young diagrams — CHARLEN LEE