Fall 2018, Representation Theory of Compact Lie groups

Course Syllabus: Click Here.

Weekly Schedule: Announcements about homework, special lectures, etc. will be posted here.

  • 1. Week of August 1st.

Lecture 1 (01/08): Definition of a Lie group.

Examples: GL(n,R), GL(n,C), SL(n,R), SL(n,C), PGL(n,R), PGL(n,C), O(n), U(n).

Review of the topological properties of a space being (i) compact, (ii) connected.

A fun exercise in elementary number theory.

Galois groups: compact topological groups (but not Lie groups).

Lecture 2 (02/08): More examples: O(n), SO(n), U(n), SU(n), Sp(n), Lorentz group: SO(3,1).

(For us Sp(n) will be the compact symplectic group; Notation to be compared with Sp(2n,R) or Sp(2n,C).)

Maximal compact subgroups of general linear groups.

Special Lecture on Basics of Group Theory: August 3rd, Friday, 5:40 p.m. to 7:00 p.m., Madhava Hall.

Homework 1: Click Here. (Due Date: 8th August in class. Submit exercises 4 and 10 for grading.)

  • 2. Week of August 6th.

Lecture 3 (06/08): Tangent vectors, and the tangent space at a point p of a manifold M.

Examples of tangent space computations: M = V, GL(n,R), SO(2).

Functoriality of tangent space: The Adjoint representation.

Lecture 4 (08/08): Review of Lecture 3.

Tangent Bundle of a manifold.

Vector fields on a manifold; the Lie bracket of two vector fields.

Lecture 5 (09/08): The Lie algebra of a Lie group.

The Adjoint representation of a Lie group.

Homework 2: Click Here. (Due Date: 15th August in class. Submit exercises 4 and 10 for grading.)

  • 3. Week of August 13th.

Lecture 6 (13/08): The differential of determinant is trace.

Integral curves on manifolds and one-parameter groups of a Lie group.

The exponential of a matrix.

The Lie algebras of SL(n), O(n), SO(n), U(n), SU(n).

Lecture 7 (15/08): The Lie algebras of SO(n) and U(n).

Explicit examples of exponentials of matrices.

(Note: This is a special lecture on independence day. There won't be any lecture on 16th, as I am out of town.)

Homework 3: Click here. (Due Date: 23rd August in class.)

  • 4. Week of August 20th.

Lecture 8 (20/08): The exponential map of a Lie group.

Properties: Smooth, locally diffeomorphic, and natural.

(The notes are as in Lecture 7.)

(No lecture on 22/08 being a holiday for Bakri Id.)

Lecture 9 (23/08): Classification results for: connected abelian Lie group, and a compact abelian Lie group.

Homework 4: Click here. (Due date: 29th August in class.)

  • 5. Week of August 27th.

Lecture 10 (27/08): Lie subgroups. Submanifolds. Closed subgroup theorem. A continuous homomorphism between Lie groups is smooth.

Lecture 11 (29/08): Homogeneous spaces. Quotient groups. If N is a closed normal subgroup of a Lie group G then G/N is a Lie group.

Lecture 12 (30/08): Examples: SL(2,R)/SO(2), GL(2,R)/B(2,R), and SO(n)/SO(n-1).

Homework 5: Click Here. (Due date: September 3rd in class.)

  • 6. Week of September 3rd.

Lecture 13 (03/09): Haar integrals.

Lecture 14 (05/09): Examples of Haar integrals.

Lecture 15 (06/09): Basic definitions of representation theory.

Homework 6: Click here. (Due Date: September 17th, Monday, in class.)

  • 7. Week of September 10th.

Lecture 16 (10/09): Complete reducibility theorem. Schur's Lemma.

Note: there will be no more lectures this week (I am out of town on Wednesday (12th) and Thursday (13th); but 13th is a holiday anyway).

  • 8. Week of September 17th.

Lecture 17 (17/09): Multilinear algebra.

Lecture 18 (19/09): Basic character theory; characters of direct sums, duals, tensor product, symmetric and exterior square.

Lecture 19 (20/09): Characters = degree 1 representations; commutator subgroup and abelianization; all the characters of GL(2,R).

Homework 7: Click here.

Midterm Exam: Friday, September 28th, 3:00 to 5:00 p.m., in LHC 203.

Syllabus for midterm exam: Based on Homework sets 1 through 7; and up to Lecture 19.

One A4 size paper per student with hand-written notes (on both sides) will be allowed.

Some Announcements:

Last Lecture for this course will be on November 12th.

Special lectures on these Fridays: October 5th, October 12th, October 26th, November 9th.

All these special lectures will be from 5:30 to 6:30 pm in Madhava Hall.

  • 9. Week of October 1st.

No lecture on October 3rd as I am of town.

Lecture 20 (04/10): Schur's orthogonality relations; orthonormality of characters; irreducibility criterion;

Example: the standard representation of S_4 is irreducible. (This is true for S_n.)

Lecture 21 (05/10): The character of a permutation representation; the regular representation of a finite group; Burnside's formula.

Example: The dimensions of irreducible representation of S_4 are 1, 1, 3, 3, 2.

  • 10. Week of October 8th.

Lecture 22 (08/10): Proof of Schur's orthogonality relations; Regular representation; Bases for Class functions.

Lecture 23 (10/10): Representations of G x H, Character Table of a finite group.

Homework 8: Click Here. (Due in class on Monday, October 15th.)

Lecture 24 (11/10): Character tables for S_3, S_4, A_4, S_4. Get started on Induced representations.

Lecture 25 (12/10): Induced representations and Frobenius Reciprocity.

Frobenius's paper which has induced representations and Frobenius reciprocity.

  • 11. Week of October 15th.

Lecture 26 (15/10): Mackey Theory. Example: the irreducible representations of dihedral groups.

Homework 9: Click Here. (Due in class on Wednesday, October 24th.)

Festival Break from October 17 to October 21.

  • 12. Week of October 22nd. (Representations of compact Lie groups SO(2), SU(2), U(2), SO(3).)

Lecture 27 (22/10): Characters of SO(2). Construction of irreducible representations of SU(2).

Lecture 28 (24/10): All irreducible representations of SU(2). Representations of U(2).

Lecture 29 (25/10): Irreducible representations of SO(3) on the space of harmonic homogeneous polynomials.

Lecture 30 (26/10): Completing the proofs of classification of irreducible representations of SO(2), SO(3), SU(2) and U(2).

Homework 10: Click Here. (Due in class on Wednesday, October 31st.)

  • 13. Week of October 29th. (Peter Weyl Theory for a compact topological group G.)

Lecture 31 (29/10): The structure of the space spanned by matrix coefficients M(G) inside C(G).

Lecture 32 (31/10): Statement of Peter-Weyl theory: M(G) is dense in C(G), and L^2(G) as a representation of G x G.

Lecture 33 (01/11): Consequences of Peter-Weyl theory: There is a faithful representation. Every irreducible is finite-dimensional.

Homework 11: Click Here. (Due in class on Thursday, November 8th.)

  • 14. Week of November 5th.

Lecture 34 (05/11): Clean-up Peter Weyl theory. The representation theory of the Lie algerbra sl(2).

Lecture 35 (08/11): The representation theory for a general compact Lie group via the example: U(n).

Root space decomposition, Root Systems, Weights of a representation, Highest weight.

"Highest Weight Theory": The highest weight knows everything of the irreducible representation that it parametrizes.

Weyl Character Formula. Weyl Dimension Formula.

Lecture 36 (09/11): Special Lecture by Professor Sunil Mukhi.

Title: The role of representation theory of compact Lie groups in Physics.

Venue: Madhava Hall. Time: 5:30 p.m.

Group Photograph taken on November 8th, 2018.