Undergraduate Seminar, Spring 2011 - Symmetric functions, symmetric group representations, and Hopf algebras

Symmetric functions, symmetric group representations, and Hopf algebras

What: Undergraduate Seminar on symmetric functions, symmetric group representations, and Hopf algebras
When: Tuesdays 8:05-10:05pm, starting Jan. 25, 2011
Where: Math 507

Alexander Ellis
Office: Math 408
Email: (last name)at(math.columbia.edu)
Office hours: by appointment (which is not meant to deter you!)

Our goal is to study the close connection between symmetric polynomials and the representation theory of the symmetric groups, culminating in a beautiful 1976 result of Geissinger—the construction of a natural Hopf algebra structure on the ring of symmetric polynomials (in infinitely many variables). The key insight here is that studying all the symmetric groups at once, we uncover hidden structure. Along the way, we'll have a lot of ground to cover: the basics of finite group representations, the more speciaized representation theory of symmetric groups, some symmetric polynomial theory, and the basic properties of Hopf algebras.

If we have extra time at the end (and we won't), there are many directions in which we can take this material. In 1981, Zelevinsky generalized Geissinger's approach to two other (families of) families of finite groups: the general linear groups over finite fields and wreath products of a fixed finite group with the symmetric groups. The former family leads to the commonly heard folk adage that "linear algebra over the field of one element is the combinatorics of finite sets" (which is nonsense, but nonsense which can be made precise). Another direction is Schubert calculus, in which problems of enumerative geometry (how many lines pass through four general lines in (projective) 3-space?). Yet another is the connection between the theory we'll have constructed and the general linear group (over the complex numbers), which goes by the name of Schur-Weyl duality.

The only formal prerequisites are a strong grasp of linear algebra and comfort with basic group theory (say, as in Modern Algebra I). I will give the first (and perhaps the last) talk. Other than that, all talks will be student talks. Learning how to write and deliver a mathematical talk is a specialized skill and an integral part of the course.

Course policies: There are very few requirements in this course, so each one counts for quite a lot. Here they are:

Be there on time for every talk
Give one or two talks (depending on class size)
Three days before your talk(s), email the class an abstract
In the week preceding your talk(s), give a practice talk

Items (1) and (2) are mostly self explanatory. If you will need to be late or absent from a class, let me know in advance. Only very good excuses will be accepted. Showing up late or not at all is disrusptive to the class and disrespectful to the speaker. Each week we meet for two hours, which will be split into two student talks. If there are more than 13 students then not everyone will get to give a second talk. It is in your best interests to give two talks, since just about everyone will give a much better talk their second time (and therefore get a much better grade); also, the first few talks are less fun than the later ones, so if you speak in the first few weeks you'll want to speak again.

Since some talks will inevitably fail to cover all the assigned material, the list of lectures below the schedule on this page will have to be shifted a bit or even supplemented with "slack" talks. Because of this, I'll only assign the next 2-3 weeks of talks at a time. When you claim a talk, it's your responsibility to schedule a meeting with me in which we'll go over what you will talk about. Once we've done this, you will have to (3) email a short (<200 words) abstract of your talk to the entire class, at least three days before your talk. Finally, you have to (4) give a practice talk. This is crucial! If you do not give a practice talk first, I promise you that your in-class talk won't be nearly as awesome as it could have been. I want you all to support each other by attending each other's practice talks. This is not a formal requirement, but please go to each other's practice talks. After you give yours, please email me a list of who attended.

From time to time, I may email out a few practice problems. These are optional and for your benefit only. If you're giving one of the later talks, I strongly suggest doing these so that you keep up with the material. I'm happy to go over them with you during my office hours. I will try and keep them on the easy side. Speaking of office hours, they are by appointment, preferably on Mondays or Wednesdays.

There is a lot of good material online on how to give a good mathematical talk. Here are three articles:

by Jordan Ellenberg
by Joseph Gallian
by Gian-Carlo Rota (only some parts relevant)

Texting or emailing during a talk is rude. Don't do it. Wait until between talks.

Given the timing of the seminar, you are allowed (nay, encouraged) to bring something to eat. Just no crinkly wrappers, please.

Literature: You'll need much, much less than what's below.
[Artin] M. Artin, Algebra: A good undergraduate algebra textbook. Chapter 9 covered finite group representations and section 14.3 has a proof of the Fundamental Theorem of Symmetric Polynomials.
[Bump] D. Bump, Lie Groups: A textbook on Lie groups. While we won't need Lie groups at all, sections 34-39 and 43 cover various aspects of symmetric polynomials. While some of the development is from the point of view of Schur-Weyl duality, much of it should be readable. See especially the exercises to section 37. Section 49 discusses the "philosophy of cusp forms," and in particular has some details on the representation theory of GL_n(F_q). Section 50 has some material on Schubert calculus.
[FH] W. Fulton and J. Harris, Representation Theory: A First Course: An introductory book on representation theory. The first part of the book covers most of what we need on finite groups and has the most complete account of representations of symmetric groups at an introductory level.
[Geiss] L. Geissinger, "Hopf algebras of symmetric functions and class functions": Original account of the Hopf algebra structure on the ring of symmetric polynomials. Short and readable.
[GH] P. Griffiths and J. Harris, Principles of Algebraic Geometry: Section 1.5 covers Schubert calculus on Grassmannians. Of course you have to have some command of basic algebraic geometry of complex manifolds to understand this.
[KMS] M. Khovanov, V. Mazorchuk, and C. Stroppel, "A brief review of abelian categorifications": This is way past the level of our course, but it's also one of the best introductions to a certain notion of categorification.
[Maniv] L. Manivel (trans. J. R. Swallow), Symmetric Functions, Schubert Poynomials and Degeneracy Loci: Chapter 1 is a good account of the theory of symmetric functions. Chapters 2 and 3 take this a bit further, in the direction of Schubert calculus.
[Plato] Plato, The Republic: In all seriousness, the allegory of the cave (Book VII, numbers 514-520) may as well be an allegory of categorification. The ring of symmetric functions is the mere shadow of the interplay of induction and restriction functors between categories of representations.
[Serre] J.-P. Serre, Linear Representations of Finite Groups: Serre is a master expositor. Chapters 1-3 and 5-8 along with section 9.1 cover all of the general representation theory of finite groups that we will need. When reading Serre, make sure you are filling in the details yourself as needed.
[Stan2] R. Stanley, Enumerative Combinators, vol. 2: Chapter 7 is a comprehensive account of the theory of symmetric polynomials. The book is also a treasure trove of good exercises.
[Stee] J. M. Steele, The Cauchy-Schwarz Master Class: Besides being one of my favorite math books of all time (it's a problem book on identities and inequalities), this also covers some identities satisfied by symmetric polynomials.
[Tele] C. Teleman, "Representation theory": Notes from a course on representation theory, focusing on finite groups. By the end, they even cover the symmetric group a bit. Very readable.
[Zel] A. Zelevinsky, Representations of Finite Classical Groups, A Hopf Algebra Approach: This book lays out some structure theory for a class of Hopf algebras which arise in settings such as Geissinger's construction. Then it treats several cases in this framework: symmetric groups, the groups GL_n(F_q), and wreath products of a fixed group G by the symmetric groups. The general theory adds some unappealing overhead to the proof, but the book does have the merit of generality (in particular, Zelevinsky proves a strengthened Mackey Theorem needed for his examples).

Besides the literature, Mikhail Khovanov maintains an excellent list of online resources for representation theory. See especially the headings on finite groups and on the symmetric group.

Exercises, etc.:

I have prepared some notes on graded rings. I hope they come in handy.

Here are some optional exercises. Let me know if you want help with any of them.

Talks:

1/25	Introduction		Alex Ellis
2/1	Representations of finite groups 1	[Artin], [Serre], [Tele]	Lara Avsar	Abstract
2/1	Representations of finite groups 2	[Artin], [Serre], [Tele]	Nora Diamond	Abstract
2/8	Representations of finite groups 3	[Artin], [Serre], [Tele]	Jane Kim	Abstract
2/8	Representations of finite groups 4	[Artin], [Serre], [Tele]	Hillary Duenas	Abstract
2/15	Representations of finite groups 5	[Artin], [Serre], [Tele]	Claudia Meza-Cuadra	Abstract
2/15	Representations of finite groups 6	[Artin], [Serre], [Tele]	Hechen Ren	Abstract
2/22	Representations of finite groups 7 & 8	[Artin], [Serre], [Tele]	Betsy Carroll & Alessandra Gotbaum	Abstract
3/1	Representations of the symmetric group 1	[Bump], [FH]	Jeff Irwin	Abstract
3/1	Representations of the symmetric group 2	[Bump], [FH]	Esther Wolff	Abstract
3/8	Representations of the symmetric group 3	[Bump], [FH]	Alex Frouman	Abstract
3/8	Representations of the symmetric group 4	[Bump], [FH]	Sherin George	Abstract
3/15	SPRING BREAK
3/22	Symmetric functions and graded rings	[Stan2], [Bump], [Artin]	Vanck Zhu	Abstract
3/22	Elementary and complete symmetric functions	[Stan2], [Bump]	Taylor Owens	Abstract
3/29	TBD	[Stan2], [Bump]	Evan Biederstedt	Abstract
3/29	TBD	[Stan2], [Bump]	Paul Lerner	Abstract
4/5	Schur functions 1	[Stan2], [Bump]	Claudia Meza-Cuadra	Abstract
4/5	Schur functions 2	[Stan2], [Bump]	Tasos Moulinos	Abstract
4/12	CLASS CANCELLED
4/19	Schur functions and antisymmetric functions		Alex Frouman	Abstract
4/19	Schur functions and the Jacobi-Trudi identity		Hillary Duenas	Abstract
4/26	Frobenius's approach to the symmetric group		Nora Diamond	Abstract
4/26	Hopf algebras		Jane Kim	Abstract
5/3	The Hopf algebras of symmetric functions and symmetric group representations		Esther Wolff	Abstract
5/3	The Mackety theorem and the bialgebra axiom		Paul Lerner	Abstract

Abstracts:

Lara Avsar, 2/1: The first is the basic definition of a representation and how we will be using that definition this semester. If you want to start thinking about the basic notion of what a representation is, begin thinking of general similarities between objects in mathematics. A few properties and examples will follow the definition tomorrow.

I will also be discussing Schur's Lemma. Schur's Lemma is an important result in representation discussion. It deals with irreducible representations, isomorphisms, etc. I will go over the details tomorrow, but basically I will show that, by using the kernel and the image, we must have a specific result accordingly. It deals with the maps between irreducible representations.

I will also be discussing the complete reducibility theorem. This theorem states that every finite dimensional representation is completely reducible. This is also a segway into the discussion around the idea that if there is a finite abelian group, all its finite dimensional irreducibles are 1-dimensional.
Nora Diamond, 2/1: I will discuss characters and (some of) their properties, Maschke's Theorem (that every finite dimensional representation of a finite group is the direct sum of irreducible representations), and the relationship between conjugacy classes and irreducible representations of a group. This is mostly covered in chapter 9 of Artin's text.
Jane Kim, 2/8: I will be going over examples of representations-- namely the trivial representation, regular representation and its properties, dual representation (including a brief discussion on finite dimensional dual vector spaces), and the representation of S_3. I will introduce tensor products for vector spaces and representations, and complete the proof for the orthogonality relations from the previous talk.
Hillary Duenas, 2/8: I will talk about the relationship between the number of irreducible representations and the number of conjugacy classes. More specifically, I will prove the theorem that states if G is a finite group, then the number of irreducible representation of G equals the number of conjugacy classes of G. I will then go on to define and give examples for characters, in particular using S_3, the symmetric group of 3 elements. I will then start to explain the concept of a character table. The character table will be discussed in greater detail in the next week of lectures.
Claudia Meza-Cuadra, 2/15: I will explain Tensor products of vector spaces and of representations and how these relate to characters. In particular, I will show that the character of the tensor product of two representations is equal to the product of the character of each representation. I will also describe what the direct sum and dual representation do to characters.

Then, I will further explore character tables, explaining how to interpret and calculate them and why they are particularly useful. In particular, I will show that it is possible to decompose a representation into irreducible representations using characters.
Hechen Ren, 2/15: On Tuesday we shall see a handful of powerful tools useful for calculating the character table. In particular, I will show the sum of the dimensions of the irreducible representations squared equals the order of G; a character is irreducible if its product with itself gives one; how to compute multiplicities with characters; characters evaluated at the identity give the dimension of the representation; dimensions of irreducible representations divide the order of G; if V is irreducible and W is 1-dimensional, then the tensor product of V and W is irreducible; how to get the last row of a character table. And I will wrap up with the example of S_4, the symmetric group of 4 elements.
Betsy Carroll & Alessandra Gotbaum, 2/22: We will discuss restricted and induced representations and their implications on certain properties of groups, such as character, conjugacy classes, and dimension. Restriction and induction are essentially reverse operations of one another: restriction as an operation forms a representation of a subgroup from a representation of the whole group, whereas induction forms a representation of the whole group when a representation of a subgroup is given. In discussing the details of these operations, we will cover how the character is affected in each case, and will also cover the Frobenius Reciprocity theorem, which supports the "reverse" nature of the relationship between restriction and induction. Lastly, we'll introduce idempotents. An idempotent is an operator x such that x^2=x. We will discuss how, in looking at idempotents, we can yield subrepresentations.
Jeffrey Irwin, 3/1: I will begin by quickly giving an overview of representations and what we should keep firmly in mind going forward. Afterwards, I will restate Frobenius reciprocity, which Alessandra and Betsy introduced last week, and provide a proof for it. Following this, I will introduce the group algebra and some of its properties. Also, I will discuss idempotents and how idempotents in the group algebra will yield a representation. Finally, I will give a relationship between two representations obtained by idempotents in the group algebra.
Esther Wolff, 3/1: I will discuss representations of the Symmetric groups (Sn). I will introduce Young diagrams and show how every conjugacy class in Sn gives rise to a Young diagram and a corresponding Young subgroup. We will define lexicographic (dictionary) ordering on Young diagrams and show how to take the transpose of a Young diagram. We will then use these Young subgroups to draw string diagrams of partitioned, permutation groups. I will define a Specht module and then finally, we will be introduced to an exciting theorem regarding the relationship between Induced irreps and conjugacy classes in Sn.
Alex Frouman, 3/8: I will prove the theorem about the Hom set between induced representations on Young subgroups. I will first review the various propositions from last week that are essential to the proof. Then I will introduce cosets and minimal coset representatives and the same for double cosets. Using minimal representatives from double cosets, I will then prove the theorem. We will do several examples of calculations and also prove a corollary about conjugacy classes and irreps of Sn if there is time after the proof.
Sherin George, 3/8: I start my talk by computing some induced representations from S_3 to S_4 first using Frobenius reciprocity and then a theorem using Young diagrams, which makes the computation much simpler. I will then go over what a Young lattice is and then state a corollary of the theorem giving us a statement about the dimension of L_lambda. We will then see L_(n) = trivial, L_(1, 1, ..., 1) = sign, and L_(n-1,1) = fundamental. I will also define what hooks on Young diagrams are and we will go over the hook length formula and do an example.
Vanck Zhu, 3/15: My talk will center on the ring of symmetric polynomials. We will begin with a brief review of rings and graded rings, and explore how this applies to the symmetric polynomials. We will then look into monomial symmetric functions and prove that the monomial symmetric functions form a basis for the ring of symmetric polynomials. If time allows, we will take a look at the elementary symmetric polynomials and prove that they also form a basis for the ring of symmetric polynomials.
Taylor Owens, 3/15: I'm going to pick up where Vanck leaves off, and if he can't cover elementary symmetric polynomials, I'll explain them and prove that they form a basis for the ring of symmetric polynomials. I'll then move on to the main topic of my talk, which is the complete symmetric functions. I'll explain the generating function for complete symmetric functions and show the symmetry between the elementary symmetric functions and the complete symmetric functions. I'll then move on to remind everyone of inner products and show that an inner product on two dual bases of a vector space determines that Vector space. Finally, if i have time, I'll prove that the inner product is symmetric.
Evan Biederstedt, 3/29: We'll begin with a rather reckless review of the material from Vanck's and Taylor's talks: symmetric functions. I'll introduce the concepts of the fundamental types of symmetric functions i.e. elementary symmetric functions, monomial symmetric functions, and complete homoegenous symmetric functions. By tackling explicit examples and through direct calculations (with much peer participation!), we once more see how these basic building blocks of symmetric functions relate to one another. I will subsequently introduce the concept of the power sum symmetric function and repeat as above, while working through proofs in Stanley's Chapter 7 (e.g. products of power sum functions are a generating set over the rationals).
Paul Lerner, 3/29: I'll be talking generally about the scalar product as it pertains to the ring of symmetric polynomials, building off of the fundamental types of symmetric functions that Evan introduces. First, I'll give a brief refresher on inner products and dual bases, which you may remember from your linear algebra course. Then, I'll define the scalar product as we'll use it and give some results of it. We'll see that the scalar product is symmetric and look at a lemma that will let us verify if two bases of Λ are dual bases. Then we'll see how the power sum symmetric series form an orthogonal basis of Λ and check that the scalar product is positive definite. If I have time, we'll also check that the involution ω is an isometry.
Claudia Meza-Cuadra, 4/5: I will be introducing the fifth basis of the symmetric group, The Schur Functions. I start off with a definition of Schur functions and a brief explanation, using examples, of the main concepts necessary to work with them, including Semi-Standard Young Tableaus, Shape and Content. Then I will generalize the definition of Schur Functions to Skew Schur Functions and prove that these are symmetric. I will also show how it is possible to write the Schur functions in terms of the monomial basis and finally, show that the Schur functions make up a basis for the symmetric group themselves.
Tasos Moulinos, 4/5: We will be covering the Schurr functions, which form a fifth basis for the ring of Symmetric functions. As we will see, the schurr functions have certain nice properties; they can therefore be considered as the fundamental basis for the ring of Symmetric functions. For one thing they provide an integral basis for the ring of symmetric polynomials in the integers. Furthermore, they are an orthonormal basis, which I will prove in the beginning of my talk. I will follow that with the computation of certain inner products (of the various bases elements we have studied up to now) thus highlighting some of their differences (this will also give us a chance to recap some of the important results from the previous few talks). Following this, I will show how the involution w defined in the last talk acts on the Schurr functions. I will then give an alternate definition of the schurr functions and (time permitting) prove that the object of the definition is indeed a Schurr function.
Alex Frouman, 4/19: I will begin by proving the relation stated last time between skew-symmetric functions and Schur functions. This result allows us to extend what we know to skew Schur functions. To explain this extension I will introduce the idea of a horizontal strip that shows how we can compute skew-symmetric. We can then compute skew Schur functions from Schur function.
Hillary Duenas, 4/19: I will state, prove and give examples of the Jacobi-Trudi identity. Specifically this will allow us to express s(lambda) as a determinant whose entries consist of the basis h(lambda). I will then define the Frobenius characteristic map and interpret Z(lambda) in terms of the size of the conjugacy class lambda at S_n. If time permits, I will discuss important examples of mathematics in Borges.
Nora Diamond, 4/26: I will prove the Frobenius characteristic map to be an isometry and an isomorphism. I will then show how we can generate the character table for a given Sn from the coefficients of a symmetric polynomial and consider the example of S₃.
Jane Kim, 4/26: I will introduce the notions of a coalgebra, bialgebra, and Hopf algebra, then provide examples. In the discussion, I will emphasize the diagrammatic conceptualizations of these objects.
Esther Wolff, 5/3: We will discuss the ring of symmetric functions as a graded and commutative algebra: We will continue our discussion from last week and will relate it to Representation theory by defining the terms: comultiplication, counit, cocommutivity (mutivity;) and coassociativity in terms of the class functions on Sn. We will then look at some examples, perform some computations and end with the statement of Mackey's Theory which shows how comultiplication is in fact an algebra homomorphism.
Paul Lerner, 5/3: We will cover Mackey's Restriction Formula, and use it to finish the proof of comultiplication as an algebra homomorphism. We'll then see via the Frobenius characteristic map that there is a correspondence between the coproduct on the symmetric polynomials and the coproduct on the class functions.