Department of Mathematics Syllabus

This syllabus is advisory only. For details on a particular instructor's syllabus (including books), consult the instructor's course page. For a list of what courses are being taught each quarter, refer to the Courses page.

MAT 246: Algebraic Combinatorics
Approved: 2009-03-01, Anne Schilling and Jesus De Loera

Units/Lecture:

Fall, alternate years; 4 units; lecture/extensive problem solving

Suggested Textbook: (actual textbook varies by instructor; check your instructor)
Enumerative Combinatorics, Vol. II, R. P. Stanley ($140). See supplemental texts below.
Search by ISBN on Amazon: 978-0-521-78987-5

Prerequisites:

MAT 245 or consent of instructor

Course Description:

Algebraic and geometric aspects of combinatorics. The use of structures such as groups, polytopes, rings, and simplicial complexes to solve combinatorial problems.

Suggested Schedule:

Lectures Sections Topics/Comments


Depending on the instructor, different emphasis may be given to the various topics.
1 Chapter 7.1 Ring of symmetric functions
2 Chapter 7.2 Partitions and their orderings
3 Chapters 7.3 - 7.7 Various bases of the ring of symmetric functions
4 Chapter 7.9 Scalar product
5 Chapter 7.10 Combinatorial definition of the Schur functions
6 Chapter 7.11 RSK algorithm
7 Chapter 7.12 Cauchy identity
8 Chapter 7.14 Dual RSK
9 Chapter 7.15 Classical definition of Schur functions
10 Chapter 7.15 Littlewood-Richardson coefficients and Pieri rule
11 Chapter 7.16 Jacobi-Trudi identity
12 Appendix A1.1, Sagan 3.4 Knuth Relations
13 Appendix A1.2, Sagan 3.7 Jeu de Taquin
14 Chapter 7.17 Murnaghan-Nakayama rule
15 Chapter 7.18 Characters of symmetric group


Further topics (as time allows)

The following lectures are Jesus De Loera's recommendation for the second half of the course. The text is "Combinatorics and Commutative Algebra", by R. P. Stanley.

Lectures Sections Topics/Comments
16-20 Chapter 0 Basic commutative and homological algebra
21 Chapter 2 Simplicial complexes and face lattices
22-23 Chapter 2 F-vectors and h-vectors, Dehn Sommerville Equations
24-25 Chapter 2 Face ring of a simplicial complex
26-27 Chapter 3 Two special examples: simplicial polytopes; matroid complexes
28 Chapter 3 Gorenstein face rings
29 Chapter 3 Toric varieties and g-vectors
30 Chapter 3 A proof of the Upper Bound Theorem

Additional Notes:

"The Symmetric Group: Representations, Combinatorial Algorithms and Symmetric Functions," by Bruce Sagan ($55). "Combinatorics and Commutative Algebra," by R. P. Stanley ($80).