FSF3702 Algebraic Combinatorics 7.5 credits
Content and learning outcomes
Course contents
An important part of the course is formed by the theory of symmetric functions. This is a classic topic in algebra, however the theory turns out to be of a mainly combinatorial character. The ring of symmetric functions has a basis consisting of the Schur functions. These are generating functions for so called Young tableaux.
On the combinatorial side the course will cover several topics from classical enumerative combinatorics. This concerns in the first instance partitions, permutations, plane partitions and tableaux, where some of the highlights are the hook-length formula for enumerating Young tableaux, MacMahon's enumeration formula for plane partitions, the Robinson-Schensted-Knuth correspondence between permutations (and more generally, nonnegative integer matrices) and pairs of tableaux, jeu de taquin, the theory of monotone subsequences, enumeration using non-crossing lattice paths, etc.
Intended learning outcomes
After passing the course the students will understand, and are able to apply, algebraic methods in combinatorial mathematics.
Literature and preparations
Specific prerequisites
A Master degree including at least 30 university credits (hp) in in Mathematics. Basic courses in algebra and combinatorics.
Recommended prerequisites
Equipment
Literature
The indicated chapters of the following books:
- William Fulton, Young Tableaux, Cambridge Univ. Press, 1997. [Part 1]
- Donald E. Knuth, The Art of Computer Programming, Vol.3/Sorting and Searching, Addison-Wesley, 1973. [Chapter 5.1]
- Ian G. Macdonald, Symmetric functions and Hall polynomials (Second Edition), Oxford Univ. Press, 1995. [Chapter 1]
- Bruce E. Sagan, The Symmetric Group (Second Edition), Springer, 2001. [Chapters 3 and 4]
Examination and completion
If the course is discontinued, students may request to be examined during the following two academic years.
Grading scale
Examination
- INL1 - Assignment, 7.5 credits, grading scale: P, F
Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.
The examiner may apply another examination format when re-examining individual students.
Homework, possibly combined with some seminar/oral assignment.
Other requirements for final grade
Approved assignment/presentation.
Opportunity to complete the requirements via supplementary examination
Opportunity to raise an approved grade via renewed examination
Examiner
Ethical approach
- All members of a group are responsible for the group's work.
- In any assessment, every student shall honestly disclose any help received and sources used.
- In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.