Math 701: Introduction to the symmetric group algebra, Spring 2024
Professor: Darij Grinberg

Organization

Classes:
Classes are over now! (Was: MWF 1:00 PM -- 1:50 PM in One Drexel Plaza, GL46.)
Office hours:
Monday 2--3 PM in Korman Center 263. Saturday 1--2 PM on https://drexel.zoom.us/j/2350700617. Also by appointment.
Text:
Lecture notes (source code). Work in progress.
Blackboard:
https://learn.dcollege.net/ultra/courses/_358806_1/cl/outline.
Gradescope:
Not in this course. Just send homework to darij.grinberg@drexel.edu.
Instructor email:
darij.grinberg@drexel.edu

Course description

A survey of the symmetric group algebra and various families of elements living therein: random-to-top and random-to-random shuffles, Young-Jucys-Murphy elements, cycle sums, the Murphy and Young seminormal bases and others. Representations of the symmetric group will also be discussed, as well as their connection to symmetric functions.

See my talk for a teaser.

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. Knowledge of representation theory is not needed -- we will construct the relevant representations with our bare hands.

Course materials

Recommended:
Other:

Course calendar

Homework:
  • By June 2nd at 11:59 PM, make sure to submit at least a first draft of your homework (aiming at 30 points or more).
  • The final deadline is June 10th at 11:59 PM. No more work (new or corrected) will be accepted after that.
  • Note: You can submit multiple revisions until the deadline. By submitting some work early, you can get feedback ahead of the deadline, thus allowing you to correct mistakes or add extra solutions. (For this and other reasons, I recommend typesetting your solutions.)
Plan:
The following is HIGHLY TENTATIVE. Major changes likely.
  • A first look: definitions and teasers.
  • Young-Jucys-Murphy elements: commutativity and cycle sums.
  • Cycles and the center of the symmetric group algebra.
  • Study of the random-to-top shuffles.
  • The descent algebra (following Wildon) and its denizens.
  • Riffle shuffles.
  • The one-sided cycle shuffles.
  • The Murphy elements.
  • Young tableaux and Young symmetrizers.
  • Algebraic intermezzo: Modules and representations; the Maschke theorem.
  • Examples of representations of Sn.
  • Specht modules: definitions and basic properties.
  • Standard tableaux and Garnir relations.
  • The basis theorem for Specht modules.
  • Semistandard tableaux and semistandard homomorphisms.
  • Specht modules in the Murphy basis.
  • How random-to-top and one-sided cycle shuffles act on Specht modules.
  • James's Speng modules, following Ceccherini-Silberstein et al.
  • The Littlewood-Richardson rule for skew shapes.
  • Characters.
  • Symmetric polynomials and functions.
  • Random-to-random elements: commutativity.
  • The Gaudin Bethe subalgebra: commutativity.
  • The Young seminormal form, following Rutherford.
  • Eigenvalues of the Young-Jucys-Murphy elements.
  • The hook length formula.

Grading and policies

Grading matrix:
  • 100%: homework.
    Homework problems can be found all over the notes (search for "Exercise").
    The boxed number at the beginning of the exercise is the number of experience points you gain for this exercise. It is roughly proportional to the difficulty of the exercise (somewhat reduced if the exercise is tangential or likely to be known from prerequisite courses). A typical homework exercise ranges from 2 to 5 experience points, but there will be both easier and harder ones in the notes.
    For 100% course percentage, you have to gain at least 50 experience points through the entire quarter. 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).
    Submit solutions to darij.grinberg@drexel.edu.
Grade scale:
These numbers are tentative and subject to change:
  • A+, A, A-: (80%, 100%] (that is, (40, 50] experience points).
  • B+, B, B-: (60%, 80%] (that is, (30, 40] experience points).
  • C+, C, C-: (40%, 60%] (that is, (20, 30] experience points).
  • D+, D, D-: (20%, 40%] (that is, (10, 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 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 math.stackexchange before the due date!)
  • Late homework will not be accepted.
  • Solutions have to be submitted electronically by email in a format I can read (PDF, TeX or plain text if it works; no doc/docx!). If you submit a PDF, make sure that it is readable and does not go over the margins. If there are problems with submission, send your work to me by email for good measure.
Expected outcomes:
The students will have a familiarity with the group algebras of symmetric groups, including their Artin-Wedderburn decomposition in characteristic 0, and the basic combinatorics of Young tableaux relevant to it, such as the hook-length formula. They will have seen the classical representation theory of symmetric groups from the viewpoint of Specht modules, and in particular the Littlewood-Richardson rule for decomposing a skew Specht module into straight ones. They will have also have studied some remarkable families of elements in these group algebras, such as the Young-Jucys-Murphy elements, the somewhere-to-below shuffles, and the random-to-random shuffles, as well as the descent algebra and its properties.

Further literature

General:

Other resources

University policies:
Disability resources:

Back to Darij Grinberg's teaching page.