Combinatorics in Representation Theory

Jan-Apr 2011 (9:45-11:10am, Tuesdays and Thursdays)

Brief description

The goal of this course is to understand the role played by combinatorics in resolving the problem of computing the characters of classes of finite groups. The main example will be the representation theory of general linear groups over finite fields.

Recommended reading

  1. Symmetric functions and Hall polynomials, by I. G. MacDonald, Oxford University Press, 1979.
  2. Permutations, matrices, and generalized Young tableaux, by D. E. Knuth, Pacific J. Math. 34 1970, 709-727.
  3. Harish-Chandra homomorphisms for p-adic groups, by R. Howe with the collaboration of A. Moy, AMS 1985.

Homework

Homework guidelines

List of assignments (arranged by due date)

January: 11, 13, 18, 20, 25, 27.
February: 3, 8, 15, 24.
March: 8, Midsem exam (9th March), 26.

Topics covered

4th January: Basic concepts of representation theory (representations, modules, simplicity, semi-simplicity, group algebras).
6th January: Basic concepts of representation theory continued (characters and their properties).
11th January: Numerical determination of character table, some uses of characters, true and virtual characters, induced characters.
13th January: Permutation representations and their intertwiners, Grassmannian representations, Gelfand's trick.
18th January: Decomposition of Grassmannian representations, description of intertwiners between induced representations.
20th January: Flags and their relative positions, intertwiner network of flag representations.
25th January: Knuth's theorem. Classification of representations of Sn.
27th January: Knuth's algorithm.
1st February: Enumerating conjugacy classes in GLn(Fq).
3rd February: Types of classes, Block Jordan canonical form.
8th February: Twisting by characters, derived group of GLn(Fq).
10th February: Vector bundles, induced representations, geometric intertwiners.
15th February: Intertwiners between flag representations and their twists by the sign character for Sn.
17th February: Vλ⊗ε≅Vλ′.
24th February: Classification of irreducible representations of Sn using dual RSK.
1st March: Symmetric polynomials, RSK correspondences, and irreducible characters of Sn.
3rd March: The classical definition of Schur polynomials. Frobenius character formula.
8th March: Young symmetrizers. The characteristic map.
10th March: Skew-tableaux, skew-Schur functions.
17th March: Character of a polynomial representation of GLn(C). Weyl's unitarian trick (lecture by Raghavan).
22nd March: Specialization to n variables. Fundamental property of skew-Schur functions. Littlewood-Richardson coefficients. Pieri's rule.
24th March: The Jacobi-Trudi identity.
5th April: Representations of GL2(Fq)- Harish Chandra induction and cuspidality.
7th April: Dimensions of cuspidal representations of GL2(Fq). Cuspidality for GLn(Fq). Philosophy of cusp forms.

Course notes

These are scans of the rough notes that I use while lecturing. Using statements in these notes without understanding them is not a good idea, since finer points of hypotheses and definitions are not always given. Nor is any attempt made to correct minor (or sometimes even major) errors.
Basic concepts of representation theory (pages 1-11).
Numerical determination of character table (page 11½).
Why it helps to know characters; true vs. virtual characters (pages 12-14).
Induced representations, permutation representations and their endomorphism algebras (pages 15-20).
Decomposition of Grassmannian reps; intertwiners of induced reps (pages 20½-25).
Intertwiner network of flag representations, determination of decomposition for n=3,4 (pages 26-34).
Tables for S4 (pages 35-38).
Knuth's theorem and representations of Sn (pages 39-44).
Conjugacy classes in GLn(Fq).
Types of classes, Block Jordan canonical form (pages 46-49).
Twisting, derived group (pages 50-51).
Geometric theory of intertwiners (pages 52-56).
Twisting representations of Sn by the sign character (pages 57-63).
Dual RSK correspondence and classification of representations of Sn (pages 64-66).
Symmetric polynomials and characters of Sn (pages 67-75).
Specialization to n variables (pages 75 1/3-75 2/3).
Classical Schur functions. Frobenius character formula (pages 76-80).
Young symmetrizers; the characteristic map (pages 81-85).
Skew tableaux, skew-Schur functions, Pieri rule, Jacobi-Trudi identity (pages 87-92).
GL2(Fq), Harish Chandra induction and dimensions of cuspidal representations (pages 93-98).
Philosophy of cusp forms for GLn(Fq) (pages 99-105).

Grading scheme

hw: 50%, midsem: 20%, final exam: 30%.
The following are the intervals for IMSc students:
>80% - A
>70% - B
>60% - C
>50% - D