Math 390. Algebraic Combinatorics
Lectures
Sep.06: Introduction. Oddtown. Notes.
Sep.08: Glossary of graph theoretic terms. The adjacency, incidence, and degree matrices of a graph. A matrix identity. Notes.
Sep.11: class cancelled.
Sep.13: Directed (multi)graphs. Statement of the Matrix Tree Theorem (MTT). Worksheet 1. Notes.
Sep.15: Proof of the MTT, Part 1. Notes.
Sep.18: Proof of the MTT, Part 2. Notes.
Sep.20: Applications of the MTT (using some linear algebra). Notes.
Sep.27: Counting walks using the adjacency matrix. Notes.
Sep.29: Basic definitions and examples of partially ordered sets (posets). Ideals and filters. Notes.
Oct.02: Chains and antichains. Antichains vs. ideals. Mirsky's theorem. Notes. Worksheet 2.
Oct.04: Finishing the proof of Mirsky's theorem. Discussion on Worksheet 2. Notes.
Oct.06: Finishing the proof of Dilworth's theorem. The Erdős--Szekeres theorem. Notes.
Oct.09: Dual and product posets. Meets, join, and lattices. Notes.
Oct.11: Proofs on lattices. The distributive properties. Group work on Worksheet 3. Notes. Worksheet 3.
Oct.13: Comments on Worksheet 3. Set-enriched posets. The easy half of the fundamental theorem of finite distributive lattices. Notes.
Oct.23: Join and meet of the empty set. Join-irreducible elements in lattices. Notes.
Oct.25. Finishing the proof of the fundemental theorem of finite distributive lattices. Worksheet 4.
Oct.27: Observations about Young's lattice. Linear extensions. Notes.
Oct.30: Sperner posets. Symmetric chain decompositions. Notes. Worksheet 5.
Nov.01: Continuing Worksheet 5. A generalization of Sperner's theorem: products of chains have SCDs and are hence Sperner.
Nov.03: Second approach for proving Sperner's theoerm, via order matchings. Linearizations and up/down operators. Notes.
Nov.06: Order matchings via injective and order-raising linear maps. Finishing the second proof of Sperner's theorem. Notes. Worksheet 6.
Nov.08. Differential posets. A consequence of the Leibniz rule. Notes.
Nov.10: Worksheet on an inductive proof. Group theory review. Notes. Worksheet 7.
Nov.13: The orbit-stabilizer theorem. Burnside's orbit-counting lemma. Notes.
Nov.14: (make-up class) A glimpse at the Robinson--Schensted algorithm. Worksheet 8.
Nov.15: Proof and applications of Burnside's lemma. Notes. Worksheet 9.
Nov.17: Going between actions, homomorphisms, representations and modules of groups. Notes.
Nov.20: Algebras, group algebras, and representations of algebras. Notes. Worksheet 10.
Nov.21: (make-up class) Zeus gives an introduction to his thesis.
Nov.22: More time parsing definitions and working on Worksheet 10. Examples constructions of group representations. Notes.
Nov.27: Module homomorphisms. Permutation modules of symmetric groups corresponding to partitions. Notes. Worksheet 11.
Nov.29: The dominance and lexicographic order on partitions. More practice on modules. Notes.
Dec.01: Construction of Specht modules. Notes.
Dec.04: Cyclicity of Specht modules. Computation of examples (in worksheet). Notes. Worksheet 12.
Dec.06: Examples from Worksheet 12. A basis for Specht modules. Notes.
Dec.08: The dominance order on tabloids. Towards simplicity of Specht modules: an S_n-bilinear form. Notes.
Dec.11: The sign lemma and the submodule theorem. Specht modules are simple over fields of characteristic zero. Notes.
Dec.13: The Specht modules of partitions of n form a complete, irredundant list of simple modules for S_n over a field a characteristic 0. Notes.