Math 3951: Undergraduate Seminar (Topics in Representation Theory) (Fall 2022)

This is an undergraduate seminar, where students will give most of the talks. The two topics, conducted in parallel, are combinatorial species and category O, roughly in proportion to student interest.

• When: Wednesday 8PM - 10PM ET (except the week of 09/12-09/16, when it is 09/16 12PM - 2PM ET)
• Where: room 507 (except the week of 09/12-09/16, when it is room 528)
• References (see Canvas for pdfs):
  [B] Bergeron, Labelle, Leroux, Combinatorial Species and Tree-like Structures
  [H] Humphreys, Representations of Complex Semisimple Lie Algebras in the BGG Category O
  [K] Kirillov, Introduction to Lie Groups and Lie Algebras
  [W] Wilf, Generatingfunctionology
• The notes from the seminar are not here, but here.

Schedule

Week 1

Talk 1 (Fan Zhou)

Introduction/Organization, notes here.
I will sell the topics to you. I will quickly introduce species and demonstrate their power. Then I will describe how the stage is set for category O. Everything will be rather sketchy.

Talk 2 (Fan Zhou)

continued

Week 2

Talk 3 (Chris Wang)

The Calculus of Generating Functions, Part 1, notes here.
The use of generating functions as an acting clothesline to hang up a sequence of unknown numbers to display and consider further. More specifically, interpreting and understanding members of a sequence through the sum of a power series, observing the coefficients as a given sequence we attempt to find and solve when a straightforward formula isn't apparent. Further, the basic manipulation of formal power series.

Talk 4 (Imanol Echevarria)

The Calculus of Generating Functions, Part 2, notes here.

This lecture will cover the calculus of exponential generating functions with some cool examples.

Week 3

Talk 5 (Aswath Suryanarayanan)

Basic Notions in Species, notes here.

n/a

Talk 6 (Nathan Raghavan)

Addition and Multiplication of Species, notes here.

n/a

Week 4

Talk 7 (Fan Zhou)

Summary/Recollection/Cleaning up loose ends.

Some of the previous talks didn't cover everything I was looking for and I suspect people are falling behind, so I will reinforce some ideas to delay the point of no return.

Talk 8 (Arjun Kudinoor)

Differentiation and Composition, notes here.

Last time, we learnt about the addition and multiplication of species. Today, we will define and investigate the composition and differentiation of species. We will use species composition and differentiation to derive the exponential generating series, type generating series, and cycle index series representations for a myriad of species - like those of endomorphisms on finite sets, cyclic permutations, and partitions of finite sets.

Week 5

Talk 9 (Beth Wang)

Category Theory Interlude, notes here.

n/a

Talk 10 (Rimas Chacar-Palubinskas)

Pointing and Cartesian Product, notes here.

n/a

Week 6

Talk 11 (Ashley Waller)

Example: apply Mobius inversion to compute cycle index series of Cyc, notes here..

n/a

Talk 12 (Ashton Daniel)

Examples: Misc., notes here.

n/a