Math 701: Symmetric functions, Fall 2026
Professor: Darij Grinberg
Organization
Course description
A survey of symmetric and quasisymmetric functions including their combinatorics and algebra. Some contemporary research will be discussed.
Level: graduate.
Prerequisites: a good understanding of rings and modules (as provided, e.g., by Math 332). Some familiarity with enumerative combinatorics (Math 222) and algebraic combinatorics (Math 531) will be helpful but not strictly required. Tensor products might be used in some sections, but they will be defined.
Course materials
- Recommended:
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- My Introduction to Algebraic Combinatorics. We will start with Chapter 7 and insert some other sections as necessary. The notes are detailed but don't go deep in any direction.
- Darij Grinberg, Victor Reiner, Hopf Algebras in Combinatorics (version with solutions). We will (probably) not define Hopf algebras, but we will cover much of Chapters 2, 5 and 6 in these notes.
- Mark Wildon, An involutive introduction to symmetric functions. Unofficial errata. Quick introduction to some of the main ideas (particularly the combinatorial ones), written in the laconic British style.
- Nicholas Loehr, Combinatorics, Chapman & Hall, 2nd edition 2018. Unofficial errata. Chapters 9-10 cover symmetric functions in a combinatorial fashion (too combinatorial, if you ask me). We might follow the treatment of k-abaci and k-cores given here.
- Steven Sam, Math 740 Symmetric Functions. Unofficial errata. Quick and terse but useful.
- Alistair Savage, Symmetric Functions. Unique for its Chapter 5, but we will unlikely cover that.
- Amritanshu Prasad, An introduction to Schur polynomials, Graduate J. Math. 4 (2019), 62--84: A short introduction.
- Eric S. Egge, An Introduction to Symmetric Functions and Their Combinatorics, AMS 2019. Unofficial errata. Nice and elementary introduction, though not exactly going deep.
- Bruce Sagan, The symmetric group, 2nd edition, Springer 2001. Official errata. Chapter 5 is devoted to symmetric functions, though not quite my way of introducing them.
- Martin Aigner, A Course in Enumeration, Springer 2007: Another textbook with some symmetric function theory (symmetric functions are Chapter 8).
- Bruce Sagan, Combinatorics: The Art of Counting, AMS 2020. Official errata. Has chapters about symmetric and quasisymmetric functions, focusing on their uses in enumeration.
- Ian G. Macdonald, Symmetric functions and Hall polynomials, Oxford Science 1995. Long and austere monograph with a lot of advanced algebraic material; we will unlikely get past Chapter I.
- Richard P. Stanley, Enumerative combinatorics, vol. 1 and 2, Cambridge University Press 2011 and 2024. Encyclopedic reference on algebraic combinatorics; Chapter 7 of volume 2 is about symmetric and quasisymmetric functions. Too terse and distracting to read from front to back, but a rich source of interesting results.
- Richard P. Stanley, A symmetric function generalization of the chromatic polynomial of a graph, 1995. This paper started the long romance between symmetric functions and graph theory.
- Darij Grinberg, Double posets and the antipode of QSym, arXiv:1509.08355. Includes detailed proofs of various fundamental properties of quasisymmetric functions (see the long version).
- Pavel Galashin, Darij Grinberg and Gaku Liu, Refined dual stable Grothendieck polynomials and generalized Bender-Knuth involutions, arXiv:1509.03803. Introduces these power series and shows their symmetry.
- Darij Grinberg, Petrie symmetric functions, arXiv:2004.11194. A rather natural family of symmetric functions with some interesting (though not very deep) properties.
- D. Laksov, A. Lascoux, P. Pragacz, and A. Thorup, The LLPT Notes, 2018. This is the old, determinantal approach to symmetric polynomials and some related objects. We will probably not touch on this except perhaps for a couple subsections.
- On harmonics and coinvariants: TBA, if we ever get to that (Jim Haglund could tell this story much better). For now, see Polynomials over Symmetric Polynomials for an introduction.
- Anthony Mendes, Jeffrey Remmel, Counting with Symmetric Functions, Springer 2015. To be honest, I have barely ever peered into this book. Maybe this course will make me take a proper look at last.
- Andrius Kulikauskas, Symmetric Functions of the Eigenvalues of a Matrix, 1993. Answers a rather natural question: what comes out when a symmetric polynomial is applied to the eigenvalues of a matrix.
- Lynne M. Butler, Subgroup lattices and symmetric functions, Memoirs of the AMS, 1994. Supposedly a readable (if dated) introduction to the Hall-Littlewood world. I'll see if I can get something out of this.
- Siddhartha Sahi, Interpolation, Integrality, and a Generalization of Macdonald’s Polynomials, 1996. Maybe...
Course calendar
Grading and policies
- Grading matrix:
- 100%: homework.
There will be two homework sets.
The first homework set consists of the exercises in Section A.6 of my Introduction to Algebraic Combinatorics, except for Exercises A.6.1.8 and A.6.3.14 (which are too close to theorems I will prove in class).
Your goal is to solve enough of them to achieve 40 experience points.
(There are 215 experience points in total to be gained here, thus a lot of choice.)
The second homework set will gradually appear through the quarter and will be due in the last week; it will give another 40 experience points.
The experience points obtainable for each exercise are shown as a boxed number at the beginning of the exercise.
This number is roughly proportional to the difficulty of the exercise (somewhat reduced if the exercise is tangential or likely to be known from prerequisite courses).
Partial credit will be given for half-correct solutions and for parts of a multipart problem.
You can (and are encouraged to) submit piecemeal to gradually collect points. This will be much easier for me than grading 20-page long submissions on the last day of class.
You are also welcome to submit preliminary versions to get early feedback and then correct your solutions from the feedback (unless you ask for spoilers/solutions, in which case you forfeit any further points on the given problem).
- Grade scale:
- These numbers are tentative and subject to change:
- A+, A, A-: (60, 80] experience points.
- B+, B, B-: (40, 60] experience points.
- C+, C, C-: (20, 40] experience points.
- D+, D, D-: (0, 20] experience points.
- Homework policy:
- Collaboration and reading is allowed, but you have to write solutions in your own words and acknowledge all sources that you used.
- Asking AI tools or outsiders (anyone apart from classmates and Drexel employees) for help with the problems is not allowed. (In particular, you cannot post homework as questions on the internet before the due date!)
- Late homework will not be accepted.
- Solutions have to be typeset (not handwritten) and submitted electronically via Canvas in a format I can read (PDF, TeX or plain text if it works; no doc/docx!). If there are problems with submission, send your work to me by email for good measure.
- Expected outcomes:
- The students will have gained a working familiarity with symmetric and quasisymmetric functions, including some of the related combinatorics and algebra.
Other resources
- University policies:
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- Disability resources:
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Back to Darij Grinberg's teaching page.